Yours, Etc.: Origins and Uses of 8 Sign-Offs
1926, American English, possibly a variant of English dialectal fainaigue "to cheat or renege" (at cards), of unknown origin. Liberman says finagle is from figgle, phonetic variant of fiddle "fidget about," frequentative of fig. Related: Finagled; finagling.
[1920s+; origin unknown; perhaps related to British dialect finegue, ''to evade,'' or fainague, ''to renege'']