With just one sink you have to use it for everything - dishes, hand washing, teeth brushing, etc. (We clean it a lot with those fabulous Clorox Wipes!) We don't leave the dish scrubber in the sink, but with such limited space, we can't leave it on the counter, either.
I found this sponge holder (at Walmart, I think) and we took the suction cups off of it and replaced them with 3M removable hooks that swing so that the whole thing travels inside the sink, but doesn't actually touch it. Amazon has them, too.
We just move our faucet (which moves in all kinds of directions) out of the way and close the lid. It's worked perfectly!