Interesting question, but some history on Mt. Rushmore and the purpose of the carvings would help us to provide the truest analogy for Boston's Mt. Rushmore of sportr.
A few Mt Rushmore facts:
1. Mt. Rushmore construction began in 1927 and was completed in 1941.
2. President Coolidge who presided when the decision to erect the sculpture was made insisted that in addition to Washington, that 2 republicans and 1 democrat be selected.
3. The presidents were ultimately selected by the sculptor who used the criteria of commemorating America's 1st 150 years by honoring presidents with key roles in preserving the Republic and expanding its territory. Thus, Washington (obvious), Jefferson (Lousiana Purchase; founder role), T. Roosevelt (Conservationist; Panama canal construction that reduced freight route fm San Fran to New York by 8,000 miles), and Lincoln (of course, for preserving the union through the civil war).
Therefore, the Boston sports analogy would be to select those who preserved or expanded sports (or their particular sport) in Boston:
My choices;
Red Auerbach (constructed the greatest sports dynasty ever; built 3 generations of champions); Orr (single-handedly changed his entire sport, and put the Bruins on the Boston map in the 70's); Bill Belichick (changed the face of the NE Patriots); Cy Young (pitched baseball into local and national prominence)
Difficult omissions: Robert Kraft, Bill Russell, Larry Bird