Prince of Persia

The development of the SNES port seems to be quite interesting as its graphics are most definitely traced from the Apple II release, hinted at by the leftovers found in the game files and there is reason the believe that code too was lifted off of the Apple II, but aside from this, what influence did Mechner have over this port? The design of the new levels is almost as good as the original ones, if not better at times and there are many ideas included in the port which seem like something Mechner himself might have put in the game if he had the time to. I'm just curious.

Based on the info available, we can speculate that this port indeed used both the code and graphics from Apple II as there are leftovers of both within the game. We could ask Mechner himself on Twitter if he had a direct hand or contribution on this port, but like pretty much almost every other official/licensed port, the answer is that he probably didn't besides the Apple II code itself that was written pretty much by him.
Also, a good chunk of the original levels are there but just expanded or with some modifications in the puzzle-solving.

It was developed by the same company that made the PC-98 port of Prince of Persia (which is the origin of the high-res graphics used for the Mac version). Mechner wrote this in his journal about the PC-98 release which makes it sound like he had no involved in the development of that port:

Saw the new NEC 9801 Prince from Japan. It’s beautiful. It blew me
away. What a great feeling. As Lance said: “It’s like seeing the movie
version of a book you wrote.”

I found this journal entry from Feb 1992 which definitely makes it sound he also had no involvement in the development of the SNES port:

Went to Activision to see the Super Nintendo version of Prince. Wow!
It was like a brand new game. For the first time I felt firsthand what
it’s really like to play Prince of Persia, when you’re not the author and
don’t already know by rote what’s lurking around every corner.