birthday – день рождения

birthday (n.)
late 14c., from Old English byrddæg, "anniversary celebration of someone's birth" (at first usually a king or saint); see birth (n.) + day. Meaning "day on which one is born" is from 1570s. Birthnight is attested from 1620s.

birth (n.) (рождение, роды, начало, источник; из скандинавских языков; из протогерманского «gaburthis»).
early 13c., from a Scandinavian source, cf. Old Norse *byrðr (replacing cognate Old English gebyrd "birth, descent, race; offspring; nature; fate"), from Proto-Germanic *gaburthis (cf. Old Frisian berd, Old Saxon giburd, Dutch geboorte, Old High German giburt, German geburt, Gothic gabaurþs), from PIE *bhrto past participle of root *bher- (1) "to carry; to bear children" (cf. Sanskrit bhrtih "a bringing, maintenance," Latin fors, genitive fortis "chance;" see bear (v.)). Suffix -th is for "process" (as in bath, death). Meaning "parentage, lineage, extraction" (revived from Old English) is from mid-13c. Birth control is from 1914; birth rate from 1859. Birth certificate is from 1842.
birth (v.)
mid-13c., from birth (n.). Related: Birthed; birthing.

day (n.)
Old English dæg "day," also "lifetime," from Proto-Germanic *dagaz (cf. Old Saxon, Middle Dutch, Dutch dag, Old Frisian dei, Old High German tag, German Tag, Old Norse dagr, Gothic dags), from PIE *dhegh-.
PIE корень *bhrto- из *bher – нести, вынашивать.
Это русское «беру» (брать) - «бремя» - «беременная». БРАТЬ – БРТ – замена «т» - «th». BRTH – birth. И, глагол, по-видимому, был раньше.
Ну, а «day» - дею, день. Время для действия, а не для сна.