c. 1400, " a book that lies permanently in some specified place" (especially a large copy of a breviary in a church), noun from leggen "to place, lay" (see lay (v.)). Perhaps formed on the model of a Dutch word; the -er seems to indicate "that which has been."
Commercial sense of "book of accounts" is first attested 1580s, short for ledger-book (1550s). Ledger (adj.) "remaining in a place, permanent, stationary" is attested from 1540s; compare ledger-bait "fishing bait made to stay in one place" (1650s).
The surname, however, is via the Normans, from St. Leger, a 7c. bishop whose memory was popular in France and Normandy. The name is Germanic, *Leodegar, literally "people-spear."