Ideally there would be an IC which takes in a composite video signal (in its many forms) and drives a variety of different LCDs (especially the one I'm planning on using). It seems as though they exist. You can buy LCD driver boards on eBay which take in composite and drive an LCD, and it looks like a single chip solution. The downside is that I haven't been able to find a datasheet, or a reliable source for any of these chips. If I recall correctly, Analog devices also has an IC which takes in composite and drives and LCD. The (major) problem with this is that it is designed as an integrated solution for TV's and is massive (in terms of area and price).

The best solution seems to be using an IC dedicated to converting the video input into a digital format (search "ntsc decoder ic"), and using an FPGA to drive the LCD from the information received. From what information I've gathered so far, the typical output resolution of the decoders is 720x525 (NTSC) or 720x625 (PAL). This pixel data would need to be scaled to the appropriate size for the display. Storing 3 lines of video data just about fills the block RAM for the FPGA dev board I'm using. This should be just enough of a buffer to scale the pixel data and output to the LCD. It would be a huge pain to have to use external RAM, so I really really hope this works out.

I've ordered an LCD from Adafruit to play around with. I haven't been able to find a better deal on an LCD screen, so it's in the design for now.