What is the origin of the "-th" suffix? What is the linguistic term for the meaning it adds to words?

I was teaching my young nephew some math the other day, and from discussing the typical sort of word problems he's encountering in class, I noticed that the "-th" suffix adds a distinct meaning to adjectives. For example:

• If a ship is long, it has length.
• If a woman is wide, she has width.
• If a person is strong, he possesses strength.
• If what I say is true, I'm speaking truth.
• A lumbering panda moving slow is full of sloth.

Now, I've learned some linguistics from English L&U, and I'm guessing this "-th" suffix is an affix that changes adjectives into nouns. My questions are: What exactly is this "-th" suffix adding to the meaning? Secondly, does the "-th" originate from a separate word in Old English? Lastly, is there something to say about the vowel shifts that seems to be occurring in some of the transformations (e.g., strong going to strength) that somehow fits in with the ablaut system of strong verbs/weak verbs, that I learned of from the excellent responses to my previous question?

It is, as you say, a nominalising suffix, that goes back at least to Common Germanic. It usually surfaces as "-th" in English, but as "-t" after a (historical) "-(g)h": "height", "weight", "sight", "flight".

The vowel alternations are mostly a matter of length (though I admit that the "-ong"/"-ength" alternation sounds a bit like IE ablaut). "Wide" in Old and Middle English was bisyllabic, with a long vowel, whereas "widþ" had one closed syllable and therefore a short vowel. Only the long vowel went waltzing round the mouth in the Great Vowel Shift. Thinking about it, this particular alternation may go back to a time when the IE ablaut was operating, but I'm not sure.

In general, the suffix -th that creates nouns from adjectives comes from Proto-Germanic *-iþō.

Since this was already a suffix in Proto-Germanic, it definitely was not a separate word in Old English. In fact, it seems to descend from a Proto-Indo-European suffix, the same one that is the origin of Latin -(i)tas (which shows up in English borrowed words as -(i)ty).

The vowel shifts are not related to ablaut. There are two main causes:

• umlaut, or i-affection: although the original i in the suffix was lost by the time of Old English, it left a trace by making a preceding vowel front, changing a to æ or e, o to e, and u to y (which became i later on).
• regular shortening of long vowels before consonant clusters, and inconsistent shortening of long vowels in some cases before final single consonants

Examples:

• the Old English adjective strang/strong "strong" corresponded to the noun strængþ(u)/strengþ(u) "strength," and the Old English adjective lang/long "long" corresponded to the noun lengþ(u) "length". These both show umlaut.

• In Old English, the noun corresponding to slāw "slow" was slǣwþ, with the expected umlaut, but the word later got remodeled to have the vowel of the adjective: sloth. Sloth has two pronunciations in modern English: it may have a shortened vowel, like cloth (from Old English clāþ) or it may not, like oath (from Old English āþ).

• The adjective foul is actually related to the noun filth. The latter shows umlaut and shortening: the words descend from Old English fūl and fȳlþ.

• the noun width shows shortening of the originally long vowel seen in wide.

• the adjective true corresponds to Old English triewe/treowe. (It would be more etymologically accurate if we used the spelling "trew" in modern English, but we don't because English spelling is a mess). Since the vowel in Old English (ie or eo) was front already, it didn't change in the noun triewþ/treowþ "truth".

• breadth has an interesting history. It does show umlaut, as can be seen from comparing it to the adjective broad (from Old English brād; the development of the vowel in modern English is irregular). ‎But in Old English, it actually did not have the -th suffix: the umlaut was due to a Proto-Germanic form suffixed with something like -j- or -i-, which was lost resulting in brǣdu (cognate to German Breite). The suffix -th was added to it in Middle English by analogy with words like length and width. The vowel also ended up being shortened, which is not surprising since the word ends in a consonant cluster "dth" and the Middle English vowel that corresponds to the "ea" in this word (/ɛː/) was sometimes shortened even before a single consonant (as in bread, threat, death, deaf).