How did “to wish that” come to hate the present tense in the subordinate clauses it governs, and why is it alone in this?
You asked quite a few questions. Here is an attempt at providing answers to a portion of them.
1. Is this something new or something old? Has it always worked this way in English even before the Conquest, or did we get it grafted onto us by the Norman French?
I am surprised you passed without comment OED's sense 1a(b) [with object clause with may or (formerly) present subjunctive, occasionally indicative], which is now pretty much obsolete but which includes such old but attested examples as I wish I suffer no prejudice by it (1661), I wish the house is not rob'd (1691), He is certainly bewitched: I wish the old hag upon the green has done him no mischief (1756), I wish we have not got King Stork, instead of King Log (1823). That's your answer to this particular set of questions: the highly specialized behavior of wish is relatively new. As recently as the first half of the 19th century, wish could take as complement a content clause in the present subjunctive or even indicative.
2. The first and second mystery.
I interpret these as asking what is the actual rule that can predict what sorts of finite complements to wish result in acceptable sentences.
As I said in the comments, and as others stated in their answers (a few said this after I did, though now I see that some have also said this before I did), the answer is that a finite complement of wish must convey modal remoteness. Grammatically, this means that a finite clause complement of wish must use ether the modal preterite or else the irrealis were (the latter is what some sources call the past subjunctive, but CGEL has an argument against that analysis, pp. 87-88). This is pretty much what CGEL says in various places. Thus, for instance, we need I wish she had eaten fish last night. In CGEL, the example [29iv] on p. 1003, #I wish [you passed your driving-test tomorrow], is marked (by the '#') as 'semantically or pragmatically anomalous'. CGEL says that
wish cannot be used with a 'pure' future, one where there is no present time involved: cases like this are still within the realm of hoping, so that instead of [iv] we would say I hope you pass your driving-test tomorrow.
3. Do any other verbs work in this special way?
Arguably, no. You can rephrase this as the following questions: 1. in what sorts of constructions do we find modal preterite and irrealis content clauses? 2. Are there any verbs other than wish that appear (when taking a finite clause complement) only in such constructions?
CGEL lists four kinds of constructions where we find modal preterite and irrealis content clauses (pp. 1003-1004): (a) remote conditionals, (b) complement to wish, (c) would rather/sooner/as soon (also prefer, as a marginal possibility), and (d) it be time. However, (c) allows the subjunctive (I'd rather it be sooner (source)), while (d) allows both the subjunctive (it is time he see that he has earned that right (source), it is time he submit his letter on no confidence (source)) and the present tense (It is time he enters into the twenty-first century or disappears. (source)); see also here. True, even in (c) and (d), the subjunctive and/or the present tense are much less frequently used than the modal preterite and irrealis, but they are nevertheless used at least sometimes. Thus, none of the other possibilities are as specialized as wish when it comes to what kind of finite-clause complements they can take.
TaliesinMerlin's answer seems to be the bee's knees ("business"). No etymological-historical answers here (sorry), no bounty sought. Just deeply intrigued by this verb, for the first time in my life, now you draw attention to its oddity.
Similarity to reported speech
What about this:
he said he ate fish on fridays / he said he eats fish on fridays
he said he had eaten fish yesterday / he said he ate fish yesterday
he said he would eat fish tonight / he said he will eat fish tonight
What is the name (if any) given to the mood of this "backshifted" content phrase in the first parts of these pairs? Could it be inferential? The wish construction is very similar to reported speech... but with wish it is mandatory to use the backshifted version.
Difference to reported speech
Reported speech introduces a note of doubt: just because "he said X" doesn't mean X is true.
"I wish that she would eat fish tonight", I suggest, along with the "ate" and "had eaten" versions, involves a different type of mood.
I had suggested the desiderative mood, or possibly the hortative mood (for phrases such as "I wish you would eat fish tonight"), but after reading TaliesinMerlin's answer the most likely suggestion for most cases appears to be the optative mood.
Which came first?
I.e. between reported speech and the wish construction. Is it possible that this modern wish that construction modelled itself on reported speech? Or the other way round? No idea.
TaliesinMerlin also says that these are probably "cousin constructions" which developed in parallel. This is interesting if so, since the reported speech version, by its nature, allows indicative in the content clause, or something other than indicative. Whereas the wish version abhors the indicative: only the optative (or hortative) is permitted, but other than that they mirror one another quite strikingly. It seems that our ancestors' linguistic sense knew that it was imperative for wishing to be expressed with a substitute irrealis mood of some kind as the true grammatical subjunctive fell into decline. Could we say that this construction is therefore in some sense a living fossil, or testimony to the resilience of "subjunctive feeling" in English?
When?
Also refer to TaliesinMerlin's authoritative answer for when these constructions may have entered the language, and how: 16th, 17th centuries or so... but not a neat transition, as he/she says.
This answer will describe the historical context for the verb to wish and what forms the main verb in the subordinate clause following wish can take.
Indo European, Greek, and Latin
First, Indo European languages sometimes use one of two moods to indicate a wish. (Reminder: verbal moods indicate the speaker’s intent or attitude; common moods include the indicative, the imperative, and the subjunctive.) The first is optative mood, which indicates a wish or hope. This is one of four moods that likely came from Proto-Indo-European. It is present in Ancient Greek; one gives a wish or a hope by putting the appropriate suffix on the verb, and sometimes an additional particle (adverb-like item), εἴθε, (eithe) which is glossed “would that!” (Perseus Tufts). Latin (among other languages) folds the same function into the subjunctive mood, and often accompanies its wishes with utinam (again “would that” or “if only) (Perseus Tufts) (The Latin Library).
In Latin, the verb opto (I wish) was often used to set up a subordinate clause, headed by ut, that would indicate what was wished for. The verb within the subordinate clause would take the subjunctive case or the infinitive, resembling the optative subjunctive (examples from Perseus Tufts):
(Phaëthon) optavit, ut in currum patris tolleretur,” Cic. Off. 3, 25, 94
(He wished that he were carried up in his father’s chariot) (tolleretur is in the passive imperfect subjunctive)
optavi, peteres caelestia sidera tarde,” Ov. Tr. 2, 57.
(I wished that you reached the heavenly stars late (peteres is in the active imperfect subjunctive)
“hunc videre saepe optabamus diem,” Ter. Hec. 4, 4, 29
(We often wished to see this day) (videre is in the infinitive)
The verb volo (I wish, will, want) works similarly (Perseus Tufts), as do a few other verbs of wishing, desiring, and fearing. If the verb in the subordinate clause is in the subjunctive, it will tend to be in imperfect tense (translated to past tense in English). These rules tend to be covered under either the “optative subjunctive” or the “substantival clause of purpose”; in general, expect the subjunctive, which means expecting either a past or present tense.
So even outside of English, languages have rules that are specific to wishing (the optative subjunctive) and rules that tend to govern subordinate clauses (when in a subordinate clause of purpose, use the subjunctive). So it should be little surprise that wishing works similarly in Old English.
Old English and Onward
Old English has a subjunctive and it has verbs of wishing. The Old English subjunctive is odd because there is no distinct first, second, or third person form. So for bēon (to be), the past subjunctive is wǣre in the singular and wǣren in the plural, and the present subjunctive has two forms, bēo and sīe (with –n in the plural) (Introduction to Old English by Peter S. Baker, p.67). The book notes that the subjunctive is used in clauses following verbs of desiring and commanding (p.85):
Ic wysce þæt ic wisra wære
(I wish that I were wiser) (waere is in the past subjunctive)
To compare to a similar situation, the subjunctive can also be used in indirect discourse:
Hie cwædon þæt he wære wis.
He said that he was wise. (waere is still subjunctive; “he said that he were wise” feels unusual in Modern English)
However, in a situation like indirect discourse, the indicative can also be used:
Be him awrat se witega Isaias þæt he is stefn clipiendes on westene.
Concerning him, the prophet Isaiah wrote that he is the voice of one crying in the wilderness (is is in the indicative)
Wish remains locked into the subjunctive in Old English, which only evolves into periphrastic or indicative uses over time. This is reflected in entries 1a, 1b, and 1c in the Oxford English Dictionary. I requote 1b and 1c as the most relevant:
(b) with object clause with may or (formerly) present subjunctive, occasionally indicative: expressing a desire that the event may happen or that the fact may prove to be so, and often implying some want of confidence or fear of the opposite (now commonly expressed by hope: see hope v. 3b). Also expressing a request (see 5).
(c) with object clause with past subjunctive (or indicative, e.g. was for were): expressing an unrealized or unrealizable desire (see also will v.1 46a), or in modern use sometimes a mild request (cf. 5). to wish to God: to wish intensely.
Both entries show that, over time, the use of the subjunctive decayed and was replaced by the indicative or by the use of periphrastic constructions using verbs like may or would:
(subjunctive weore) 1362 Langland Piers Plowman A. v. 92 Þenne I wussche hit weore myn.
(indicative had never known) a1561 G. Cavendish Metrical Visions (1980) 539 Therfor my frayltie I may bothe Curse and ban. Whissyng to God I had neuer knowen man.
(subjunctive were) 1578 J. Lyly Euphues f. 31 I wish my wish were as effectually ended as it is heartely looked for.
(construction with would) 1625 J. Ussher Let. in R. Parr Life J. Usher (1686) Coll. lxxiv. 315 I could wish that Mr. Lisle would take some pains in translating the Saxon Annals into our English Tongue.
The transition isn’t neat though, with the 16th and 17th centuries featuring a lot of periphrastic, subjunctive, and indicative examples in subordinate clauses following wish.
Meanwhile, 1d and 1e support the use of the infinitive after about the 14th century, and continuing thereafter. Here is one example:
c1560 A. Scott Poems (S.T.S.) xv. 39 Away I went,..Wissing all luvaris leill to haif sic chance.
All of this corresponds to the general decline of the subjunctive mood in English and the growth of other options to replace it. Middle English (a) lost verb endings and hence features that would have distinguished the subjunctive, (b) fostered periphrastic constructions with sholde, shal, wil, may, and can), (c) used the past tense indicative as a substitute for the subjunctive, as a modal preterite. These changes continued into early modern English to the present day (Eva Kovacs, “On the Development of the Subjunctive from Early Modern English to Present-Day English, p. 82).
Concluding the Mysteries
What you’re observing with wish are the trace elements of the old subjunctive system, where periphrastic constructions, past-tense verbs (substitutes for the subjunctive), or sometimes the infinitive are far more often permitted than present-tense verbs.
So for the first mystery, the present tense indicative would have always been avoided, first with the subjunctive and then with its replacements:
(x) I wish that she eats fish on Fridays.
I wish that she ate fish on Fridays. (Subjunctive-ish past tense or preterite)
I wish that she eat fish on Fridays. (Bare infinitive)
I wish that she would eat fish on Fridays. (Periphrastic modal in present)
For the second mystery, the past tense indicative would also be avoided if its subjunctive and indicative use became too ambiguous. In your example, you shift from speaking about a repetitive action (eating fish on Fridays) to a single moment in the past. In the history of English, one way to disambiguate a subjunctive sense is to go to a past indicative form. Because the verb is already in the past, taking another step into the past would lead to a periphrastic past perfect form. The periphrastic constructions appropriately conveys both tense and mood:
(x) I wish that she ate fish last night.
(x) I wish that she eat fish last night.
(x) I wish that she would eat fish last night.
I wish that she had eaten fish last night. (Periphrastic past perfect)
I wish that she would have eaten fish last night. (Periphrastic modal in past)