It was really the reverse. The idea of a suffering or defeated messiah originated in Christianity, and passed from there into Judaism (as a sort of minority opinion). My guess would be that this happened after the defeat of the bar Kochba rebellion in 135 C.E. In the Jewish version, there are two messiahs -- one who is defeated in battle (the bar Kochba figure) and one who is subsequently triumphant. This gets a mention in the Talmud, but not much more than that. That the Jewish version of this postdates the Christian version is discussed in one of Raphael Patai's books. And despite claims to the contrary concerning one manuscript, Geza Vermes, the preeminent scholar of the Dead Sea Scrolls and second temple Judaism, likewise agrees that there is no definitive evidence of the existence of this idea before the rise of Christianity.
This is confirmed, in a way, in the gospel accounts themselves. The disciples of Jesus are shocked to hear him foretell his own death. They certainly didn't seem to have a concept of a suffering messiah. If this was an idea with wide currency at the time, one wouldn't think his disciples would have expressed such surprise.