I answered this question for the local paper recently. Even most people at Mars don't know the answer to this. They, and others, mistakenly presume that both Ms stands for Mars, yet they are called M and Ms for a reason.
M&Ms were first produced in 1941 from an idea by Forrest Mars when he saw soldiers in the Spanish Civil War eating sugar-coated chocolate. The sugar coating protected the chocolate from melting and this inspired M&Ms, a chocolate that "melts in your mouth, not in your hand". The first M in M&Ms is for Forrest Mars, the inventor of M&Ms.
William Murrie was president of the Hershey Chocolate Company. His son, R. Bruce Murrie, became a partner with Forrest Mars in his project and he helped Forrest acquire and modify Hershey plant machinery to be able to produce M&Ms. The two of them initially set up the company M&M Ltd.
The second M in M&Ms stands for R. Bruce Murrie. Some sources say the second M stands for his father, William Murrie, as he had given his blessing to the partnership.
Forrest Mars eventually bumped R. Bruce Murrie out of the partnership, hence people at Mars say the Ms stand for Mars and Mars, while historians at Hershey say the Ms stand for Murrie, a well-known Hershey family ... and Mars.
Source: The Emperors of Chocolate by JG Brenner (Broadway Books, New York 1999)