Tag searching

The "regular" (default) search already searches tags (in addition to other fields--including the image title).

If you need to exclude the image titles from search, you'd have to custom edit G3 (not the search module) and the way it creates the search cache fields.

Best option is, of course, to deal with the bad image names.
You could easily use an automated script to quickly rename the images to something that doesn't conflict with your search-specific wordings.
You could also edit G3 to create the image title from some other field (maybe using a static prefix + a sequential key?).

(I'm not 100% sure if by "image name" you really mean the image title, or if you really mean the file name, but everything above applies either way.)

EDIT: Actually, I think you could do it just modifying the search module and the fields it searches. Editing the search module may be better if you're ONLY searching the tags field, but NOT if you're going to search multiple fields.

Anyway, changing the image/file names/titles is the best bet. (Kudos to @floridave for pointing out the modules below.)

Report back if you get stuck somewhere.