My sons mother and I split in November 2012 and since then has not contacted me once. She moved to another state and I have no idea where; can I just give up my rights as parent so she can go on with her life and me mine?

It sounds like you already have. What is it you think your going to avoid by giving up rights?

There is a myth that a parent can just give up their rights, but its only a myth.

Are you on the birth certificate? Or been acknowledged the legal father? Have there been any court proceedings on custody?

You cannot legally give up your rights. But practically you have. However, there is nothing to stop the mother from going after you for support in the future.