Here's a quick rundown:
Haymarket was originally Union-Friend, named for nearby streets. It was extensively rebuilt in the 1970s. (Wikipedia)
Government Center was originally Scollay (and Scollay Under for the Blue Line), named for the square above. It was obliterated by the Government Center construction in the 1960s (when the line was rerouted and the neighborhood leveled. (Wikipedia)
Hynes was originally Massachusetts, named for the Mass Ave. It then changed to Auditorium, then to Hynes Convention Center-ICA and, when the ICA moved, to Hynes Convention Center. During automated station announcements, the announcement sounds choppy at the end, as they just clipped the audio file when the station was renamed. (Wikipedia)
Prudential was originally Mechanics, named for the meeting hall which stood nearby, and changed when the Pru was built in the 1960s. (Wikipedia)
Kendall and Charles have both been modified to Kendall-MIT and Charles-MGH over the years.
Downtown Crossing was originally Washington, Winter and Summer. This is one of several Orange Line stations which has offset platforms named after multiple streets (and Washington was named after Washington Street of course). The Orange Line came first here, and the station wasn't renamed to a single transfer point until 1967, when Winter and Summer were dropped, and 1985, when it as changed to Downtown Crossing. So although things didn't change from 1944 to 1965, Tom Lehrer's Subway Song today would be out-of-date. (Wikipedia)
Washington Street was originally the only street along Boston Neck and carried the Post Road to New York. As the main street, intersecting streets changed names when they crossed Washington (which doesn't help the whole Boston is impossible to navigate thing). And because the BERy named subway stations for their intersecting streets, different platforms got different names (the Washington Street Subway had no center platforms). So, Friend-Union, Winter-Summer and …
Chinatown was originally Boylston and Essex. Boylston on the southbound side, Essex going north. Was it a problem that there was a separate Boylston station a block away? Apparently not. (Wikipedia)
State was originally Milk and State on the Orange Line and Devonshire on the Blue Line. It was, for a time, actually called State/Citizens Bank as Citizens was trying to get the whole area renamed for them, but that lapsed and the name reverted to State. (I don't think any system maps were ever displayed with this travesty of a name.) State was probably retained over other names because it is located at the old State House, in addition to being on State Street. Along with Park, it is the only main transfer station that hasn't been renamed completely. (Wikipedia)
Aquarium was originally Atlantic (for the Avenue, not the ocean).
There's a good animated GIF of the changes over time here.