Please ignore my almost two year old recommendations above - I was very wrong. Its funny now to read this, as I was so certain at the time. I've learned a lot since then (through trial and error as well as good advice), and although I thought I understood how the iMac G4 worked - I really didn't. I would not do anything I wrote above as a TMDS controller is superfluous (the actual LCD essentially contains one already) and conversion to LVDS is expensive, difficult, and in my case - potentially dangerous.
The LCD already "understands" TMDS and TMDS is the signaling that a DVI cable uses. So if you know where each wire goes, have the right DVI connector, some resistors and a power source you can make the iMac G4's LCD terminate in a DVI cable in the base that can plug into whatever you wish. The power terminates in a molex plug - so it can be plugged into any compatible power supply. And you do not have to open the LCD or the neck. I have created a very easy to follow (solder free) step by step guide for the 17" version of this. You can read it here:http://dremmeljunkie.blogspot.com/2011/08/guide-step-by-step-17-imac-g4-tmds-to.html
I have also figured a way to use the 20" (more complex power setup), but a full guide is pending. Preview it here:http://dremmeljunkie.blogspot.com/2011/11/20-imac-g4-genuine-mod-inverter-solved.html
I have a setup which uses a G4 and G5 iMac which uses an external mini, Fitting the new mini in the base may be difficult. But for those that want to resurrect their broken iMac G4s as either external monitors or all-in-ones I hope you find this useful.