Kevin,

You’re right about the origins of St. Thomas’s Third Way. In On the Power of God, a work dated after the Summa contra Gentiles but before the Summa Theologica, he explicitly attributes the argument to Ibn Sina, with developments added by Ibn Rushd. (In fact, all of the arguments used in the Five Ways are explicitly attributed to other philosophers at some point or another in St. Thomas’s work.)

You wrote: What is “Thomistic”?

Of or pertaining to the thought of St. Thomas Aquinas, a Doctor of the Church and the leading figure of the revival of Aristotelian philosophy during the High Middle Ages.

St. Thomas famously believed that, although many of the mysteries of Christian faith could not be discovered by natural reason, and had to be revealed by the grace of God, there were at least some doctrines of natural theology, and in particular, the existence of an uncaused, necessary, and perfect Creator of the visible world), which could be proven through rational demonstration, and set out his Five Ways to prove the existence of God. The third of the Five Ways, the argument from possibility and necessity, is intended to demonstrate that there must be a single necessary being — i.e. a being which could not possibly fail to exist — to explain the existence of contingent beings — i.e. beings which do exist, but could fail to exist.

Kevin’s right about the origins of the argument; in an earlier book, Aquinas explicitly attributes the development of the argument to two Muslim commentators on Aristotle — the Persian philosopher-physician Ibn Sina (known in Europe as “Avicenna”) and the Spanish-Arab philosopher Ibn Rushd (known in Europe as “Averroës”). Generally speaking, a lot of the revival of Aristotelian philosophy in Christian Europe during the High Middle Ages was deeply influenced by the work of Muslim scholars a century or two before; indeed, without the texts that Arab scholars preserved and copied, the renewed interest in classical Greek learning in Christian Europe would hardly have been possible: most of the work of Plato and Aristotle, among others, had been completely lost in Western Europe for hundreds of years, until Muslim scholars re-introduced it.

Hope this helps.