The etymology of the word “beer” is somewhat ambiguous, but there are several possible sources.
The Old English “beor” means “strong drink, beer, mead,” and is an obvious precursor to the modern “beer.” But “beor” is a word of much-disputed and ambiguous origin. (Relatedly, the Old English “breowan” means “brew.”)
The word is probably a result of a West Germanic monastic borrowing of the Latin infinitive “bibere,” which means “to drink,” and which also gave the root to the word “imbibe.”
Other scholars believe that “beer” traces back to the Proto-Germanic “beuwoz,” meaning “barley.” (Similarly, the Spanish “cerveza” is thought to link back to the same origins as the word “cereal.”)
There are several other theories, but they are not as well regarded.
For a very in-depth look at all the possibilities, along with the history of the word “ale,” visit the Zythophile blog.