Ok, well first of all I wouldn't assume that it's automatically the motherboard at fault here, regardless of what Dell tech support told you.  Normally if the motherboard goes fubar the machine won't boot up at all.

First thing I would do is reformat the hard drive and reinstall the operating system.  Odds are pretty good this will fix the problem.  You may run into a situation where the hard drive doesn't want to reformat, if such is the case then odds are pretty good is your hard drive that has gone belly up, not your motherboard.

If it turns out that Dell was indeed correct (which is unlikely, however remotely possible) and a new motherboard is necessary your best bet to find a replacement will be on Ebay.  You can't just buy any old motherboard, you need to get the correct motherboard for your make and model of laptop.  Odds are good that only a very small number of motherboards were made that will fit in your laptop correctly and have all the connections in the same place, so make certain whatever motherboard you buy is for the correct make and model for your laptop.

But like I said, odds are good dell was wrong and the motherboard is fine.