Of course I'm hardly an expert on all periods of history and it's probably just that my interests just focus on the wrong locations/eras, but it seems to me that after various Jasons of note in the Classical era, I can't call to mind any famous figures with the name until very recently.
The question of why Jason was never a popular name is hard to answer conclusively. I will say that it's Greek, and names of ancient Greek origin have never been as common in Western Europe as those of Latin or Biblical origin. Ancient Greek as a language was almost completely unknown to Western Europe until the late, late Middle Ages, so they had no access to all that literature. You'll notice that Ulysses is a much more common name throughout history than Odysseus.
As for why there was suddenly an explosion of Jasons near the end of the 20th century, I'm almost positive that it's because of the 1963 movie Jason and the Argonauts. It was one of the first great special effects extravaganzas (done by Ray Harryhausen, the pioneer of stop motion) and it was extraordinarily popular. It was like the Lord of the Rings of its decade. I think the fact that the name's popularity peaked in the '70s would support this theory.