There's no way to do this in the editor as far as I know. But you can do it manually by editing the room XML files, which is pretty straightforward.
This is the basic process, assuming you're using studio. You'll need a non-Notepad text editor such as Notepad++ or Sublime Text if you don't have one already. Open the .room.gmx file in the rooms folder that you want to change. Use the "find and replace" feature to replace the name of your old background with the new one, e.g. bgCoolTiles -> bgCoolerTiles. You can scroll down to all the <tile> elements to verify that all the the bgName properties were changed. Save the file and tab back to GM. Click yes to the "want to reload the file?" dialog if it appears. If you don't have that preference enabled, reopen the project. The room should now be changed. Yay!
If you're not using studio (grrrrrrr), get
Gmk-splitter, and you can decompose your project file, find-and-replace in the room file, and recompose it. The XML will be formatted differently but find-and-replace still works just fine. Be sure to save a backup before using Gmk-splitter just in case.
Knowing that you can edit various resource files manually can really come in handy sometimes.