weave (v.1)
Old English wefan "to weave, form by interlacing yarn," figuratively "devise, contrive, arrange" (class V strong verb; past tense wæf, past participle wefen), from Proto-Germanic *weban (cognates: Old Norse vefa, Middle Low German, Middle Dutch, Dutch weven, Old High German weban, German weben "to weave"), from PIE *webh- "to weave;" also "to move quickly" (cognates: Sanskrit ubhnati "he laces together," Persian baftan "to weave," Greek hyphe, hyphos "web," Old English webb "web").

The form of the past tense altered in Middle English from wave to wove. Extended sense of "combine into a whole" is from late 14c.; meaning "go by twisting and turning" is from 1640s. Related: Wove; woven; weaving.
weave (n.)
1580s, "something woven," from weave (v.). Meaning "method or pattern of weaving" is from 1888.
weave (v.2)
c.1200, "to move from one place to another," of uncertain origin, perhaps from weave (v.1). From early 14c. as "move to and fro;" 1590s as "move side to side." Use in boxing is from 1818. Related: Weaved; weaving.