Well, Dragonstone isn’t richer than Harrenhal, but it’s pretty rich. Likewise, Rhaegar could probably call upon other income streams: even with Aerys’ paranoia and Tywin’s frugality, a crown prince is going to get an income to keep them in sufficient splendor so as to add luster to the royal court (as opposed to being an embarrassment). Likewise, it’s possible that Elia came with a dowry or had incomes of her own that Rhaegar could call upon. And yes, Rhaegar might have borrowed based on his future incomes, it was a quite common trap that a lot of heirs found themselves falling into.

But overall, the very fact that the theory was bruited at the time suggests its plausibility – if it was evident on its face that there’s no way Rhaegar could possibly have afforded to finance the tournament, no one would have believed the rumor and it wouldn’t be given credence in the text.

And finally, Rhaegar doesn’t necessarily have to pay for the whole thing himself – he only would have had to pay for enough of the cost to make the rest affordable for the Lord of Harrenhal.