I have been playing around a little bit with this to try to reproduce your problem but I am unable to. When I select a saved search from the bottom left Saved Searches menu, and then make a change to it, on either gear icon the Save option is grayed out for me, but the Save As option is available. The thing is when I use the Save As option, even if I name the search the exact same as the one I am working with, it will create another copy of that search, with the exact same name.
I guess the best approach I could suggest with this is when you make changes to a saved search, to do a save as and then either delete the old one, or keep both and just give different names or version numbers or something like that. I have not done to much with saved searches so I learned something new today playing with that. Seems odd that there is no way to make a change to a search, but to just have to save a whole new copy of it if you make a change.