What about they go across the Himalayas and get busy in India, there's a lot of stuff to conquer there
That way they won't go to the Middle East or Europe
It would butterfly the crusades into an epic Muslim win
Himalaya option as for the route to India will.... not be a logical choice, at best. The seemingly only way to enter India by land is through Khyber Pass, and that won't be so easy as well. Also, the contemporary situation in India was rather stable under the still formidable Delhi domination, so it would've been a hard deal to confront them. I would even say that Mongols might were actually aware of those things I mentioned, hence they took the MidEast option, which was a softer cake anyway, with all the relatively easier terrain you've got to go through, more easily digestable wealth presence and easily the remotely more prestigious target, not to mention the rather sentimental motivation behind it....
That said, if the Mongols had actually chosen to invade India instead, at the best case, both sides will take a good beating, and that will certainly halt Mongol expansion in Western Asia, while the resulted changes in India will certainly be interesting as well....