It's a defence itself and it was VERY popular in 1920-s.
Just so that things don't get blown out of proportion, the Budapest defense NEVER really enjoyed a lot of popularity in top level Chess... Even in the 1920s, It's been used less than 20 times by the top ten players during the 20s through the entire decade.
It has been used a bit more extensively by lesser Hungarian players (And I think a few German ones too. And only a few of them).
Still. From time to time, against the right opponents, in the right tournament situation, even GMs use it. But it's rare. Very rare.
there are some pretty exciting known games with it (Akiba Rubinstein lost very quickly to it in the first tournament he ever faced it, for example).
Still, the opening enjoyed a certain level of popularity only among lesser players, never among the top players. And even among these, not too much...
I actually find it an interesting opening. Don't count popularity among GMs as THAT important in choosing your opening repertoir, but also realize you're playing something that has never really caught on...
3.Nh3 surely you mean the line: 1.d4 Nf6 2.c4 e5 3.dxe5 Ng4 4.e3 Nxe5 5.Nh3.
In any case, this odd looking move does seem to score well against the budapest but perhaps because budapest players dont expect to come across this line very often and it catches them off guard every once in a while.
After 5.Nh3 fritz suggests 8 moves which can reach the evaluation '=' so the line more or less aims at a middlegame struggle. White is aiming for control of d5 via Nh3-f4-d5 and Nc3-d5. Black can develop his bishop on b4 like bogo indian or fianchetto his bishop on g7 like in indian games. Interesting and natural is 5...Ng6 and white will have to come up with another plan because the black knight can take the h3 knight if it goes to f4 (black would have superior pawn structure if exf4). Black is going to exploit the fact that the pawn is on c4 and look at the d4 square.