Since no one has mentioned this yet... (and questions like this come up a lot, so maybe there should be a FAQ or something.)
The monitors inside the G3 iMacs are not "normal" Multisync monitors. They're simplified fixed-frequency units that only support a few oddball resolution/refresh rate combinations. (640x480@100-ish Hz, 800x600@86-ish, and 1024x768@70-ish.) They cannot sync to "normal" VGA rates, and they do not have standard VESA DDC lines to communicate their strangeness to a standard video card. The motherboards in these systems can support normal monitors, which is why ATX-style hacks work. It's the original wiring harness that informs the motherboard to use custom video modes on the stock monitor. Wiring up to a Mini could technically "work", but you'll need a software hack to get picture. (It might *possibly* work if you picked something close to the original mode, like 1024x768@70hz and fiddled with whatever adjustment pots the monitor might have.)
I seem to recall someone doing a hack like this with a Mini-ITX board and Linux at some point, but Linux is "easy" because it's pretty trivial to use a custom X11 modeline. (Old versions of Linux which ran on iMacs usually have an appropriate modeline present in their default XF86Config files.)
Edit: I suppose it's also worth noting that the original monitor is the most failure-prone component of an iMac G3. There's no telling how long a hack like this would last. You'd probably be safer using the guts of a high-quality stand-alone triniton monitor if you could find one in good shape that would fit.
The monitors inside the G3 iMacs are not "normal" Multisync monitors. They're simplified fixed-frequency units that only support a few oddball resolution/refresh rate combinations. (640x480@100-ish Hz, 800x600@86-ish, and 1024x768@70-ish.) They cannot sync to "normal" VGA rates, and they do not have standard VESA DDC lines to communicate their strangeness to a standard video card. The motherboards in these systems can support normal monitors, which is why ATX-style hacks work. It's the original wiring harness that informs the motherboard to use custom video modes on the stock monitor. Wiring up to a Mini could technically "work", but you'll need a software hack to get picture. (It might *possibly* work if you picked something close to the original mode, like 1024x768@70hz and fiddled with whatever adjustment pots the monitor might have.)
I seem to recall someone doing a hack like this with a Mini-ITX board and Linux at some point, but Linux is "easy" because it's pretty trivial to use a custom X11 modeline. (Old versions of Linux which ran on iMacs usually have an appropriate modeline present in their default XF86Config files.)
Edit: I suppose it's also worth noting that the original monitor is the most failure-prone component of an iMac G3. There's no telling how long a hack like this would last. You'd probably be safer using the guts of a high-quality stand-alone triniton monitor if you could find one in good shape that would fit.