A leak started a few days ago and I placed a basin underneath to catch the water. I didn't anticipate the overflow and when I checked back a couple days later there was some water damage and some blackness. There was an inch or less of warping and some peeling. Does the entire base need to be replaced or can a piece of flush wood be placed a top of it or some kind of cover to protect the surface?
The most important issue is ascertaining whether your leak has been completely repaired -- that should be done prior to doing anything to the base of your kitchen cabinet. It's not at all unusual to think you have a drip stopped only to discover it leaking again when a few days have passed. I recommend keeping an eye on the area for at least a week or two before doing anything to the cabinet base.
Once you're sure there will be no more water -- at least in the immediate future -- the permanent damage to the kitchen cabinet can be surveyed. Replacing the floor in a cabinet can be a little more difficult than it appears as the board is usually fastened very securely into the sides. Trying to remove the board if you don't know what you're doing could damage the entire unit and perhaps even the adjacent cabinets.
If the damage is severe, it would probably be a very good idea to hire a kitchen contractor for the job. They usually have the tools to cut the base neatly where it meets the sides and can install a new board that looks just like new. A good kitchen contractor can install blocking that supports the board from underneath without disturbing the other cabinets.
If the water damage isn't too bad, I have seen many sink base cabinets with a piece of sheet vinyl neatly trimmed on the floor inside. Most sink bases already have holes in the floor or rear panel for pipe access, so the vinyl dresses up what can be an unsightly interior.
Sink base cabinets have supply and waste lines in their interior and many have disposals and dishwasher lines as well. If this is the only leak you ever have in there, consider yourself very lucky. That's why I would suggest holding off on replacing the base if you feel it needs to be done. I would wait until you're ready to sell the house as you may have additional leaks between now and then, and it will need to be done again.