Since it was called Denali for a 1,000 years or so before that, why not keep the name?
The Koyukon Athabaskans who inhabit the area around the mountain have for centuries referred to the peak as Dinale or Denali. The name is based on a Koyukon word for "high" or "tall".[21] During the Russian ownership of Alaska, the common name for the mountain was Bolshaya Gora (Russian: Большая Гора, bolshaya = Russian for big; gora = Russian for mountain), which is the Russian translation of Denali.
Both words, denali and bolshaya gora have exotic sounds for a common descriptive word, "tall." The custom for naming places after people has gone on for thousands of years. The name "Rome," for instance is believed to be named, 2500 years ago, after the city's founder and first king, Romulus. Settling for a simple word for tall when there are people to venerate, is how things work.
What if the people of Illinois decided to name the peak after their native son, Abraham Lincoln? This could have happened after Alaska was purchased from the Russians in 1867 (Lincoln was assassinated in 1865). You will note that both Lincoln and McKinley were assassinated. So what if Obama decided to changed the name of Mt. Lincoln to that simple Koyukon Athabaskan name for "tall." Would that thrill just trickle down you leg?