A Good Mop
Sure, you've got one already. Check it out before you go to town on your floors; you may just spread dirt around. Is your mop a little too grimy? Are there dog hairs from the last holiday season stuck to it? Does it smell? Perhaps it's time to go ahead and get another mop head.
It might also be worth considering ditching a pattern of purchasing the cheapest mop available and invest in one with a little extra oomph. Look into microfiber mops, which use reusable microfiber cloth pads that can be soaked in a cleaning solution prior to use and dropped into a separate bin when soiled. Using several mop heads -- which can be tossed in the laundry after you're done cleaning -- means the mop head is never returned to the cleaning solution for another soaking. This prevents cross-contamination and the maddening circle of squeezing dirty mop water back into the bucket, only to spread it around again on the very floor you're trying to clean. Hospitals use them: enough said.