One of my client's motherboards has a problem with the processor socket (LGA775): a pin is broken.
Are there any solutions to this or do I have to change the motherboard?
In principle you can resolder/repair the broken pin. If this is feasible in practice depends on many things, such as
You can post a picture of the broken pin so others can have a look and give advice.
In my opinion: Unless it's a very expensive motherboard, or you can answer yes to all the questions above, a repair will probably not be practical.
You can still sell the board on Ebay or similar, noting it is defective. Someone may buy it to repair it.