I have a 20" (white) iMac with a bad logic board. The part number on the board is 820-1888-A. I'm wondering if I can upgrade it to a Core2 when I replace it?
hmm that should answer your question http://www.maclife.com/article/upgrade_your_imac_core_2_duo_processor
but 2.33 ghz are the maximum for core duo to core2 duo upgrades , hardly worth the effort and risk and money if you need a new logicboard first to even get the iMac working again, because logicboard and processor together will set you back more then a used iMac with core2 duo will cost you
if you want lion it would still be cheaper to sell the working bits and get a core2 duo iMac , we are talking about a iMac with a broken logicboard thats out of any warranty , not just processor upgrade here
the price for a logicboard and processor upgrade comes to a good $900 + and in my book thats not worth it to spend in repair costs if you could get a refurbished iMac 21.5 with i3 processor with full apple warranty for just $100 more http://www.welovemacs.com/imacintelcore1.html