Well it's located in a national park, which is controlled by the National Park Service, which under the executive branch. So I would assume that absent an act of Congress, the president has the power to tell the NPS what to call it in some way, and what the NPS calls it is what it will be generally known as. Alaskans already call it Denali and have been trying to get previous admins or congress to change the name.
EDIT: Apparently the Department of the Interior, which falls under the executive branch, has the power to do this. Interior Secretary Sally Jewell is going to issue an order Board on Geographic Names. So there's where the President can essentially decide what a mountain is called.