What is the origin of the prefix “be-”?

Question

Let's take an example.

besuchen => to visit someone

suchen => search

Answer 1

I didn't know it, but Wiktionary says it's an abbreviation of bei — which I find interesting, since there are other German prefixes that conserved the whole preposition (an, auf,...,zurück). Notice, that bei itself appears in that list (e.g. beitragen) but bei- is always a separable prefix. On the other hand be- is always an inseparable prefix. This is yields a mnemonic rule, which is always nice to have.

You might want to go directly to

Etymologisches Wörterbuch des Deuschen, dtv, 1997, ISBN 3-05-000626-9, Seite 501

or to DWDS's entry.