My answer - his mother reinherited him when he "offed" the Potters and "showed his true colors."

Which he didn't.

But everyone (including dear old Mum) thought he did, and thus, even though he was in Azkaban, he was able to inherit when she died (though he probably couldn't actually access the money).

I figure she let him back into the family because he was finally showing the ideals that she wanted. *nods*

Though that doesn't explain why he's still not on the Tapestry...maybe you can't be put back on once you're off it?