I bought a wide rubbermaid container that I use between sessions to store all my assorted small-gear (hydrometers, hoses, clamps, airlocks, all that stuff). On brew day, I put everything that I need to have sanitized in there, add a capful of Iodophor and 2.5 gallons of water. It's wide enough for the bucket lid to fit in there as well. I'll then keep that around for when I need to sanitize something quickly during the brew (like my spoon after the wort has chilled and it's been sitting out for a while).
I also have one of those adjustable strainers that I use as well to get stuff out of solution and drying, although honestly I don't stress too much about getting my gear bone-dry before using it, usually a hard couple flicks of the wrist gets most of the solution off.