Both are nice, good taste!
Yes, Jacob is often shortened to "Jake", including my son by his friends, though us in the family call him "Jacob" as we prefer it. That said, I have a friend who is absolutely never called "Jake", not once by anyone, always "Jacob". At the end of the day, I think it will come down to your son himself, and what he prefers! If he tells everyone to call him Jacob they will! I think it can be nice to have the choice between the more formal and less formal versions - is there such a choice with Jonah?