Old English becuman "happen, come about, befall," also "meet with, fall in with; arrive, approach, enter," from Proto-Germanic *bikweman (source also of Dutch bekomen, Old High German biqueman "obtain," German bekommen, Gothic biquiman). A compound of the sources of be- and come.
Meaning "change from one state of existence to another" is from 12c. Older sense preserved in what has become of it? It drove out Old English weorðan "to befall." Meaning "to look well, suit or be suitable to" is early 14c., from earlier sense of "to agree with, be fitting or proper" (early 13c.).