20" iMac G4 with mobile core i5 sandy bridge ecx board identified by lion as a macbook pro.
A mini itx will "sort of" fit. If you put it an an angle to the posts or use a metal saw to remove one of the posts, you can get it in there. Depending on your motherboard some or all of the ports may be blocked. But heat and power are an even bigger issue. The original iMac G4 was convection cooled, so the parts literally stacked on top of each other. There is less than 2" from the bottom of the drive to the bottom plate. So the optical drive can not fit. But even if you remove it, you still need the PSU,the HDD, wire, ?a graphics card, wifi, bluetooth modules etc. The only place a mini itx can fit is in the very bottom. So all that stuff would sit above it blocking the few air holes in the steel faraday cage at the top of the dome. While a low power atom or APU may work, anything else will run very hot. Ive tried it. The smaller ECX board (designed to be the size of a 3.5" drive) sits above the optical drive (replacing the original 3.5" HDD) right under the air holes and fan. Nano itx and Pico itx boards run low power processors (ex atom) but ive never seen a sandy bridge/ivy bridge variant yet, but these would be fine choices as well. For perspective this is what the 6.5" x 6.5" x 2" old mac mini (mini itx is 6.7" x 6.7" and a "low profile" atx heatsink is 75mm or 2.95" high alone):
Also remember room for ports.
Power is also a problem. Dremeljunkie.com has instructions on use of the original 17" and 20" PSUs. The ECX board uses a mobile core i/sandy bridge processor which is the same one in the sandy bridge mac minis and macbook pros/airs. There are advantages to mobile processors as it is smaller, cooler, and more power friendly (many all in ones use mobile/notebook parts). Also, the lack of a full pci-e slot isnt a huge loss as there really arent many boards that would fit and that can be powered. There is a pci-e mini slot. No atx power source will fit and pico psu's (which have external power bricks) really top out around 150 - 160 watts. The original 20" psu had 190 watts. Given that you are powering an lcd with ccfl backlights, you should target a system that consumes no more than 100 - 120 watts without peripherals. This doesn't leave much. If you are going to keep a large full atx PSU external, you may want to consider keeping the computer external so you aren't held to these limitations. The base could still be used as a USB hub or to hold an "internal" optical drive. Here is a 20" iMac G4 and 20" iMac G5 converted to external monitors via HDMI and DVI for a new generation mac mini. The iMac G4's base contains the power brick and an optical DVD writer (useful as apple didn't even make in an option for the new mini) and the iMac G5 has the iSight converted to USB.
If changing lcds just remember the 17" and 20" use the currently uncommon 16:10 ratio. The 15" I believe is 4:3.
Certainly do not want to discourage anyone and anything is possible, but I helped with some of these mods and have seen these problems arise. Consider smaller form factors, mobile processors, or even using the mac mini as an external monitor connected via DVI.