Thanks all for starting this thread because the issue of “(quod vs quid) tibi nomen est?” is something I’ve thinking about of late and I hope I’m not too late to join into the conversation.
First off, I completely agree with Carolus that the choice of “quid” makes little sense grammatically to me, yet Traupman is not the only one to make this odd choice. Wheelock uses it as well.
Qimmik was quite right too to point out that, despite any reservations we may have, Plautus seems to use quid instead of quod, with the most clear case being from the first scene of his play Amphitryon, though other plays seem to use it as well.
It would seem that that would settle the question. The problem is that it doesn’t 
First, there are a number of editions of Plautus, going back to at least the 1500’s, that say “Quod nomen tibi est?” instead of quid for the exact same line in Amphitryon quoted by Qimmik. The easiest way to verify this is just do a simple google book search for that phrase. You will find a number of examples and if you do a few spot checks, you can assure yourself that this is not an OCR mistake. It really does say “quod” and not “quid”.
Also, I found a similar instance of this with Plautus’s play Persa. The line “Quod nomen tibi est?” is used, though some texts use “quid” instead. Curiously, it seems quod is the more common choice for this passage, though I found both.
I don’t know what all this means. Is “quid” just a simple scribe error that was accidentally repeated? Was “quod” the error and it was done by well-meaning people who were trying to “correct” the text? Was there a difference in the use of “quid” in early Latin that at times it could mean the same thing as “quod” and that is why he used it? I don’t know the answer, but these are all interesting questions.
The next step is to ask if anyone else besides Plautus has used the phrase. The answer is yes, but we need to include some “late latin” texts to get a good picture.
- Terrence has one instance of using something close to the phrase, and he picked “quod”. In Phormio he writes, “Tu tibi nomen dic quod est?”
- In the book of Judges in the Vulgate, we get the question at chapter 14, verse 17: “Dixitque ad eum: Quod est tibi nomen?”
- In Nennius’s Historia Britonnum (c800’s) we get "Rex autem adolescenti ait: “Quod nomen tibi est?” Ille respondit “Ambriosus vocor.” (Of note, Bede tells the same story about 100 years earlier but avoids the issue by using “Quo nomine vocaris?” instead.)
There may be others, but those are the three I’ve found so far. The Nennius may be a bit of a stretch, but the other two were clearly written while Latin was still a spoken language.
Further we can go through the literature of the Humanists and post-Humanist periods up to about 1900. I found a number of instance of “quod”, but I was not able to find an example of “quid” (except obviously those where Plautus is being quoted).
Some instances of note because they are teaching manuals include:
- Hoole’s - Children’s Talk - English and Latin (1659’s) - “Quod est tibi nomen?” or “Qui vocaris?”
- Possellius - Familiarium Colloquiorum Libellus Graece et Latine (1681)- “Quod est tibi nomen?” or “Ede nomen tuum!”
- Adler - A Practical Grammar of the Latin Language (1858, p. 599) “Quod est tibi nomen?” or “Quinam vocaris?”
Incidentally while looking through this literature, I also noted several instances of “Qui vocaris?”, “Qui vocare?” or some equivalent also being listed. I noted this above a few times. This idiomatic phrase is attested as far back at Terrence and seems to have been commonly used. Note the analogous use of the adjectival “qui” instead of the substantive “quis,” much like the use of the adjectival “quod” instead of the substantive “quid”.
To summarize, I don’t have an easy answer. The polite thing would be to declare that either is correct, but honestly, I’m not sure that I believe that. If we look through the history of Latin, I think the overwhelming evidence is that “quod” is the word used more frequently and it has the further benefit of making sense grammatically. While we do see some evidence that Plautus used “quid”, I can not explain the deviant textual reading where some texts use “quod” and some use “quid”, nor can I say which is the earlier rendering. Further, even if we determine that “quid” was the older term, that simply makes me wonder if the difference between quid and quod was less exact during Plautus’ time and not to change my mind that “quod” is correct.
At this point, I use “quod” or I just avoid the whole business and say “Qui vocaris?” 
I’m open to other thoughts though…
Cura ut valeas,
D