The reverse lights on most cars are too dim to provide sufficient illumination when backing up. I like to see whats behind me when I am backing up and have previously installed fog lamps at the rear end of my cars, but my wife doesn't want me molesting her baby i.e. adding wiring, fuses, and relays to provide the power neccessary. My question is: can I replace the incandescent bulbs with LEDs and (presumably) get more lumens from the available current supported by the factory wiring?
You can purchase LED bulb replacements, however there are a few issues...
The main one is that most people outside the automotive lighting engineering community don't know that the back up lamps are not for you. They are a signal lamp for people / cars behind you. FMVSS108 defines and governs all automotive exterior lighting. It limits the amount, direction, and color for (your case) back up lamps. If you replace the bulbs you will likely have a non-compliant lamp. All that said, it doesn't stop people, and you will almost certainly not get a citation for having brighter back up lamps.
The other concern is that LED replacements available are mostly low performers. To do this right is rather expensive because the temperature range that these must operate in is large. Also, the reflector and lens optics are designed for a bulb filament, not a LED source... so you'll get less light out of the lamp.
So the real question you asked if if you can get a LED replacement that is brighter than your OEM bulb. Most OEM bulbs are 3156 or 3157 and produce about 32 candle power, which translates very roughly to about 400 lumens. I have never seen a LED bulb replacement produce that much output. You could do your own, but that would require you to know how to design a driver circuit with LEDs, fabricate circuit boards, and comprehend integration with automotive architecture.
For this, you'd be better off adding a second set of lamps if you really want brighter, but it would be unsightly.