Old English dægesege, from dæges eage "day's eye," because the petals open at dawn and close at dusk. (See day (n.) + eye (n.)). In Medieval Latin it was solis oculus "sun's eye." As a female proper name said to have been originally a pet form of Margaret (q.v.).
Daisy-cutter first attested 1791, originally of horses that trot with low steps; later of cricket (1889) and baseball hits that skim along the ground. Daisy-chain in the "group sex" sense is attested from 1941. Pushing up daisies "dead" is attested from 1918, but variants with the same meaning go back to 1842.