Basically everything about this question and answer irked me (besides trying to understand the distribution of "did" and "done" in this context). Sorry, Concorde, but I'm going to just go ahead and disagree with you all over the place here.
1. Everyone talks in the dialect they think is going to help them fit in to their cultural group. Dr. Baker-Smemoe at BYU even has been doing studies on the speech patterns of Mormons and Non-Mormons in Utah and, guess what! They talk differently!
2. It is great to point out that different cultural groups talk differently, use different jargon, etc., to fit in. But it is erroneous to say that one is inherently better or more proper than the other.
3. While it is possible that saying "got my hair did" started as a grammatical joke, I think it is more likely that it is based on some other speech pattern already found in that dialect. (Which is generally called, in linguistics, African American Vernacular English.) Consider these sentences, where an asterisk is something that I don't think is correct in either dialect.
1) I did my hair yesterday.
2) Yesterday my hair was did.*
3) Yesterday my hair got did.*
4) Yesterday I got my hair did.
5) Yesterday I got my truck fixed.
4 and 5 are the same construction! But one uses "did." They both may be putting the emphasis on "I" or something else; I'd hazard a guess that there is a meaning difference between "got my hair did" and "got my hair done," but I don't really have the resources to figure that one out (the meaning difference could be anything, from perfect versus progressive, or maybe something in the overall components of the sentence (maybe one can include a location and one can't, or something like that)).
4. There are definitely people in and out of the community that seriously uses "hair did" that use it as a bad grammar joke. This is pretty common when people outside a community take another community's thing, especially if the outside community thinks it is funny. That doesn't make the origin of the original use "bad grammar," though.
tl;dr: There probably is a rhyme or reason to it and it isn't "bad grammar" (whatever that even means