What Logic Board fits an iBook G3 900 mhz?

As far as I know, no. You can't change to a G4 logic board, if that's what you have in mind (at least nobody here has ever reported being able to do it, although quite a few have asked about it).

A 12-inch logic board differs in size from a 14-inch logic board.

I would think that anything 700 MHz or above would work. Some of the 500 MHz and 600 MHz iBooks had physical reset buttons. I would steer clear of those.

I'm not certain if it would work or not. I believe this is the first time I've ever read about anyone wanting to go backward.

What are the symptoms?

You may know that enough Dual USB G3 iBooks suffered from the logic board problem that Apple began the iBook Logic Board Repair Extension Program in January of 2004. Check out the FAQ page and if the symptoms look familiar, give Apple a call. If the iBook qualifies, it's a free fix.

You can find the original purchase date of the iBook, by entering its serial number into the box toward the bottom right of Apple's main support page under "About Your Support Coverage."

If it is past the three-year time limit (which it probably is), try calling Apple and ask for Customer Relations (NOT Customer Service--they can't help you) and ask them to please make an exception to the three-year limit in your case.

If the answer from Apple is absolutely, positively, "NO!" and the symptoms are those of the classic logic board failure described on the FAQ page, you may want to consider trying the shim fix suggested by John Sawyer in the Apple Discussions iBook G3 (Dual USB) Displays Forum:

[Y]ou can often fix this yourself by removing the iBook's bottom housing, and placing a shim of any sort, about 1mm to 1.5mm thick, onto the raised square on the bottom shield. I use a Scotch mounting square--you can get them in hardware stores and many grocery stores. When you reinstall the bottom case, it will press against this shim, which will press against the graphics chip, and may allow the chip to come into better contact with the logic board.
John Sawyer
CJS Macintosh Repair

Instructions for removing the lower case:

iFixit.com

Good luck.