Middle English purs, purse, from Old English pursa "little bag or pouch made of leather," especially for carrying money, from Medieval Latin bursa "leather purse" (source also of Old French borse, 12c., Modern French bourse; compare bourse), from Late Latin bursa, variant of byrsa "hide," from Greek byrsa "hide, leather." Change of b- to p- perhaps is by influence of Old English pusa, Old Norse posi "bag."
From c. 1300 as "the royal treasury;" figurative sense of "money, means, resources, funds" is from mid-14c. Meaning "sum of money collected as a prize in a race, etc.," is from 1640s. Meaning "woman's handbag" is attested by 1879. Also in Middle English "scrotum" (c. 1300).
Purse-strings, figurative for "control of money," is by early 15c. Purse-snatcher first attested 1902 (earlier purse-picker, 1540s; purse-cutter, mid-15c.; pursekerver, late 14c.). The notion of "drawn together by a thong" also is behind purse-net "bag-shaped net with a draw string," used in hunting and fishing (c. 1400). Purse-proud (1680s) was an old term for "proud of one's wealth."