You guys are thinking about it all wrong.
The sculptor of Mount Rushmore was Gutzon Borglum, and he chose the two most famous presidents, George Washington and Abraham Lincoln, and chose Thomas Jefferson because of the 1803 Louisiana Purchase (which included the land that became South Dakota). Theodore Roosevelt was suggested by United States president Calvin Coolidge.
So let's look for parallels
Two most famous Video Game characters, I'd argue Mario and Pikachu, but let's hold on to that thought.
Thomas Jefferson was chosen because he's reason they even have the land they're building on.
So, I'd argue Mario is better as a Thomas Jefferson parallel, since he's often credited with reviving the home console market, the base for which this list is built(Alternatively you could totally do R.O.B.)
Now I will replace Mario with Pac-Man as our George Washington parallel.
And finally Theodore Roosevelt was most definitely chosen because he had that cool edge to him, so will make Sonic the parallel, ironic I know.
So there's my thought out list:
Pac-Man
Mario
Sonic
Pikachu