I've been curious about this recently. I think there's a second question here, too, because I'm betting the Arabs got that name because the place they lived was called Arabia (I presume by the Romans). If so, why did the Romans call that area Arabia?
Arabs were first mentioned in both Biblical and Assyrian texts of the ninth to the fifth centuries BC where they appear as nomadic pastoralists inhabiting the Syrian desert. The fact that the name begins to be used by both cultures during the same period suggests that "Arab" was how these pastoralists designated themselves. What its original significance was we do not know, but it came to be synonymous with desert-dweller and a nomadic way of life in the texts of settled peoples.
Source: Arabia and the Arabs. From the Bronze Age to the coming of Islam, by Robert G. Hoyland.