I currently have a 20-inch Early 2008 iMac (iMac8,1), with the iMac display and an external LG display. The graphics chip is an ATI Radeon HD 2400, which, as far as I can tell from the specs, only supports two monitors. Short of getting a new computer (which appears as if it would support three monitors), what are my options for getting a third display to work?
You can purchase a USB to DVI adaptor. There are heaps of them on the market
This will allow you to keep your existing setup and power another monitor from the USB port.
Just make sure that you purchase one that is Mac compatible as a lot of them on th market are not.
I use a Matrox DualHead2Go (http://www.matrox.com/graphics/en/products/gxm/dh2go/) to drive two external monitors from my iMac (roughly the same vintage as yours, I think). It works really well.
The Matrox PowerDesk software that comes with it lets both the Mac and Windows running under Parallels treat them as two separate monitors, even though they're really configured as one big virtual monitor. No such luck with Linux running under Parallels, though.