How did Islam come to be an abrahamic religion? It makes sense to me that Christianity is, because Jesus was Jewish, as were his earliest followers. But the prophet Mohammed was not Jewish, nor were his early followers. So where did the idea originate that Muslims are descendants of Ishmael?

I'm slightly confused by the difference between your first and second question. I think you could make a very strong and very plausible argument that Islam would be an Abrahamic faith on the basis of its tenets regardless of the connection with Abrahams progeny, in the same way that I think you could argue that Mormonism is an "Abrahamic faith" even though Joseph Smith, as far as I'm aware, claims no connection to the house of Abraham.

That the Arabs, at least the Arabs of the Hejaz, were the descendants of Ishmael as far as I'm aware predates the advent of Islam. It's readily accepted as part of the genealogy of Muhammad's tribe Quraysh in the biography of the prophet. It's also mentioned in contemporary non-Islamic sources at the time of the conquests by non-Arabs.

So the Bishop Sebeos, writing in 660 about a period 30 years earlier writes of Jewish refugees from Edessa that:

"They set out into the desert and came to Arabia, among the children of Ishmael."

According to Crone's Hagarism (from which I'm quoting) the idea of Arab Ishmaelites is far older than that, appearing 800 years before the Advent of Islam in the Hebrew Book of Jubilees:

A charter for an Arab religion ofAbraham (Jshmaelite and Keturid), including mono- theism, circumcision according to the covenant, and some ethico-legal prescrip- tions, appears in Jubilees

But here we're venturing very far afield of my area of study indeed.