to bigwolf216:
I used to know someone who got a GC and then moved out of the US. Her GC expired and a few years later she found a lawyer who was willing to file another petition for her. I don't know anything about the process of giving up your GC, but based on my acquaintance's experience, it does seem possible to "give up" GC and try getting it again later. Haven't talked to her for a while so I don't know details tho....