They did not start off with a tradition of surnames but had it culturally imposed after subjugation by the English. Prior to that they used a patronymic system (ap Llewellyn, for example). So they didn't start off with a particularly large variety. Uptake wasn't immediate or widespread, so by the time it became a cultural norm to have a surname, a handful of names had spread pretty far. Add to that a small starting population to begin with. The problem worsens with every generation as surnames are removed from the pool - only an unbroken male line passes on the name, and very very few new ones are ever coined. The effect holds worldwide, but it's most obvious in cultures that didn't have a lot of surnames to begin with. This includes Vietnam (see: Nguyen) and Wales, among others.

/u/N64_Chalmers points in the right direction. Specific to the issue of Jones and patronymic devices:

The Welsh indicated "son of" with two approaches. "ap" is a cognate of the Scottish and Irish "Mac/Mc" - the "m" dropped off, and the "k" sound appears as "p" in Welsh. "Ap" like "Mac/Mc" means "son of."

The Welsh also used the "s" patronymic device. With this "Williams" - another common Welsh name, means "son of William." John was an extremely common name in Wales, so with the Welsh accent, this become "Jones" - "son of John." And that's why there are so many Joneses in Wales.