Since save games are named by the last mission you've completed, it's probably not possible to rename them, but you should be able to back them up, delete them, and start over fresh.
First, turn off the Steam Cloud Sync for SRTT. (right click the game in your library, properties, Updates tab, uncheck "Enable Steam Cloud Sync...")
Next, Navigate to your save games. (DriveLetter:/Program Files/ Steam/Userdata/YourUniqueSteamID/55230/Remote) (if you are on 64-bit, Program files will be Program Files (X86))
Create a new Folder named Backup or whatever you'd like.
Copy All files in the remote folder to the backup folder.
Start the game.
Use the Load-Game interface to delete the save game files you do not want anymore.
Exit the game.
Turn Steam Cloud Sync back on.
Start the game.
Exit the game and verify in your library that it is syncing to the cloud.
Start the game and enjoy your "new start"
At any point during these last few steps, you may get a message telling you that the cloud is out of sync, and it will ask you what version to use. Choose Local.