Ana Marie Smith

We have the same situation. My spouse's Brazilian passport is in his name prior to marriage. All other identity documents (Green Card, MN Drivers License) are in his married name.
 
We book his ticket in the name as shown in his Brazilian passport and we bring a certified copy of our marriage license on the trip along with his green card and it works fine.
For boarding the plane to Brazil and upon arrival, use your Brazilian passport. To board the plane for your return trip, show your Green Card and a copy of your marriage license to document why there is a difference between the name on your green card and the name on your ticket.  
 

Agreed - no waiver should be needed based on what was stated. But his odds at getting a tourist visa would be fairly slim. Only way to know for sure is to apply. Got $160 to spare? Go for it.
No need for an immigration attorney IMO...cheaper to just apply and find out. If a waiver were to be needed, only way to get one would be via recommendation by the CO (you can't file one on your own for a non-immigrant visa...barring rare exceptions like a K-1).

It doesn’t hurt to apply but I would consult an immigration attorney in this matter that has experience with these issues. 

No he doesn’t need any waiver. Since it’s been 11 years since he last departed the U.S. his 10 year bar of unlawfully presence has been lifted, however; because of his past immigration violation getting a non-immigrant visa will be an uphill battle. He will need strong ties in Brazil to convince the consular officer that he intends to leave after his visit and not circumvent U.S. immigration law by overstaying again. Nothing hurts to try though but as mentioned it won’t be easy. 

if he is a american born U.S. citizen then birth certificate and passport would suffice.