The word Allah was simply an Arabic word referring to the One God of Christianity (and, obviously, of Judaism before it). Mohammed probably came from a pagan tribe that had many deities, but most of North Africa was Christian (or Jewish) at that time and had been for so for centuries. When Mohammed was creating his syncretist cult, he obviously drew upon the Christian/Jewish concept of one God (Allah, in Arabic). He didn’t use the name of one of his tribe’s deities, which he could have done.
This also gave his cult a little more scope, and when he launched his attacks on Christians (who already used the word Allah or some variant thereof), it confused them and at the same time made it easier for them to bow to superior military force and “convert” to his new religion.
I had read a interesting article several years ago about how polytheistic Mecca was before Mohammad. The site of their sacred pilgrimage was a pagan shrine which honored all the Gods equally including Jesus. The people of Mecca thought Mohammad was a crazy person which supports the case for trusting first impressions.