What is the origin of "pretty" as slang for "somewhat"?
Solution 1:
As FumbleFingers says, this is a pretty old use, going back to Early Modern English:
Pretty (pri•ti) adv. Forms: see prec. [The adj. in adverbial use.]
1. To a considerable extent, considerably; in a fair or moderate degree, fairly, moderately, tolerably; [...]
1598 FLORIO Dict. Ep. Ded. 3 Boccace is prettie hard, yet understood: Petrarche harder but explained. —OED 1
(I offer this OED 1 citation rather than the earlier one reproduced by FumbleFingers because I am not convinced that the use in the earlier citation is adverbial.)
Pretty has a complicated history, as you may see from the OED 1 entry which immediately precedes the link above: from cunning, artful to clever, skilful to ‘a general epithet of admiration or appreciation... fine, pleasing, proper’, and so forth, leading to the main modern sense ‘having beauty without majesty or stateliness’; but the origin of this adverbial use clearly lies in OED 1 sense 5 for the adjective:
5. Considerable in number, quantity, or extent, as in a pretty deal, while, way, etc.; also a pretty many = a good many; ...
OED 1 gives citations for this use dating from c. 1485 down to 1861, but marks it as ‘Now arch. or dial.’
I am surprised you did not find this in any dictionary; I found it in the first three online dictionaries I checked, Collins, Oxford, and Merriam-Webster. All three post definitions for the adverb after those for the adjective.
Solution 2:
Perhaps it has to do with French influence after the Norman conquest from the expression "près de" (near) since it means "close to" or "right up to"? I have NO evidence of this. Does anyone? This just came to mind.
Solution 3:
I think it goes back much further, to the Latin 'pretiosus' meaning pricy, valuable, expensive - hence its early adjectival use in the phrase 'a pretty penny'. Pretiosus comes from the Latin 'pretium' meaning a price. Smith's Latin Dictionary has pretiosus right back in Cicero in the first century BC, and points out that 'pretium' goes right back to a Sanscrit rooot. Similarly the word 'dear' means both 'beloved' and 'expensive'. Possibly its meaning changed once the use of money became normalised, and this brought in the developed meaning Please prove me wrong, someone! it would be good to know if the earliest Early English usages were in areas where the Romans were in control, for example.
Solution 4:
Regarding the phrase "pretty please", I believe that the origin comes from a variation of the request : "I pray thee, please." "I pray thee "morphed into the single word: "prithee". Prithee is a very old contraction that was used in Shakespeare. Thus "prithee please" then became "pretty please". This is a personal conjecture, but one that rings true.