This is my theory. I don't have the skills to do this myself, but I think I know enough to at least conceptualize it in my head.

The Search button needs to be modified so that instead of just going to the Search page, it also passes along a variable. The variable would get the forumid of whatever forum you were in when you clicked the button, and pass it to the Search page, possibly through the URL.

The Search page would be modified so that instead of a drop-down selection box, it is a list of check-boxes, one for each forum, with each forum name.

Inside the Search page, with each check-box, there should be a way to check if the variable passed from when you clicked the Search button matches the forumid of the forum associated with the check-box, and if it does, set it to pre-checked.

I hope that makes sense and helps you!