You're correct that the lighting is probably the problem here. Camera phones can take some amazing photos with proper light.
Cheap is my favorite approach to lighting. Pick up some clamp lights (cheapest) or some shop lights (little more expensive, but brighter). Both can be positioned easily. If you're going to use more than one light source (which you should), make sure they are the same color temperature (don't mix and match halogen, CFL, etc).
The advantage of a constant light source is that it doesn't require you to sync a flash of any type. The disadvantage is that the light is significantly less than dedicated flashes, so a longer shutter time is required.
If you want to step it up, you can go the strobist route. Find some old flashes that can be triggered by another flash. An SB-26 is very popular for this. You can set these flashes up anywhere as long as there is line-of-site to the camera. By putting them on slave mode, they'll flash when they see a camera flash.
Advantage of this route is much more control over your lighting with a plethora of light modifier options, temperature consistency, and being able to overpower other light sources. Disadvantage is that you're looking at around $100-150/flash and you'll need a camera with a strobe flash (as opposed to a camera phone).