Maybe a mechanic. It was a humble, low-education kind of job, but still a step up from working the fields. Actually, I guess it could still be construction.

(And FWIW, the original Aramaic word applies to both stone and wood workers, and there weren't a lot of trees in the Nazareth area)

Sorry for taking your comment way more seriously than of needs to be:

I'd expect Modern Jesus' dad would be a welder, which is a well paid and respected field in our time that builds things while still being very blue collar.

What's a more interesting thought is about Mary's uncle. There's conspiracy theories about Jesus that stem from the potential relationship between Jesus and his granduncle. You see, Mary's uncle was a Trader, primarily bringing tin from Britannia (modern day Great Britain) to the heart of Rome - he also founded the first Christian church outside of Palestine (coincidentally in the British Isles). Its thought that Jesus apprenticed under this uncle for a long period of time and through that saw much of the "known world". It also appears to be the origin of the myth that Jesus didn't actually ascend to heaven after resurrection, but instead dipped to Britain (leading to my favorite line from A Knight's Tale: "The Pope may be French, but Jesus Christ was English").

Most of that last part is besides the point. Chances are, that guy would equate to a tech billionaire, and has a result be the main proselytizer one behalf of a modern Jesus figure.