False. The word Allah is the Arabic cognate with the Hebrew Eloah and Aramaic Eloha/Elah/Aloha. Arabic, Hebrew, and Aramaic are three languages which belong to the Semitic language family, other Semitic languages include Ugaritic, Akkadian, and Aramean, though these are long extinct. The generic word for "god" in these languages is 'lu (Ugaritic), ilu (Akkadian), el (Hebrew), ilah (Arabic), 'l (Aramaic), etc. In Hebrew we can say Eloah, or more specifically Ha-Eloah (the God), the definite article in Hebrew is the prefix ha-, whereas in Arabic the definite article is the prefix al-, thus al-ilah, or allah. Allah, is therefore literally translated "the God", however in English we almost never include the definite article and so we treat it instead as a proper noun: God. We do the same thing when translating the Greek hos theos, we don't say "the God" we just translate it as "God".
Before Muhammad Arabic-speaking Jews and Christians were already praying to Allah, for the same reason that German-speaking Jews and Christians pray to Gott, Japanese-speaking Jews and Christians pray to Kami-Sama, and English-speaking Jews and Christians pray to God.
This "Allah is the name of a pagan god" argument is objectively and demonstrably false. It has no basis in history, linguistic, or fact; it is an entirely fabricated claim which is intended to attack Islam but, in reality, it simply demonstrates ignorance and results in an attack on all Arabic-speaking people the world over regardless of religion. Ultimately, it just makes the one making the argument look foolish and ignorant.
-CryptoLutheran