My son came in US using my last name in his passport and green card,last year his passport was renewed and last name was changed using my husbands last name.But my son's green card is still with my last name.Question is... We are planning to go home in philippines but I'm worried that this will cause a problem with the discrepancy.we also applied for a citizenship but it will take 6-8 months.can my son still travel?

You should request name change in green card as soon as he has his new passport. File I-90.
In case you can't wait, you must have any proof of name change to submit to immigration officer at port of entry.