Well, of course, you absolutely cannot publish the student's work under your own name, so take that off the table right away. You can explain to the student that a fundamental principle of academic ethics is that one absolutely does not take credit for the work of another, and that doing so would jeopardize your entire career. So whatever the resolution of this situation is, it can't be that.
Ethically, the student could publish under a pseudonym, but I don't think it is a great idea. If the work really is that significant, researchers in the field are going to want to talk to the author: contact him to ask questions about the paper, invite him to conferences and seminars, suggest collaborations. In principle he could refuse it all, or decline to give any contact information, but this would ultimately limit the impact of his work, and it seems that it's important to him that his work should have an impact. Moreover, I think using a pseudonym would actually divert the attention away from the work and toward the mysterious author, and when his identity eventually is discovered (as I think it must), the result would only be more of the unwanted "circus clown" treatment.
Now, I think I can understand the student's fear. I think many people in fields like math and physics have had the experience of laypeople saying "I don't have any understanding of what you do, but you must be really smart." It may be meant as complimentary, but it can be very frustrating - there's an implication of "You are different from me, I cannot relate to you, and I don't want to try". It's emotionally uncomfortable. So for a person who not only works in a specialized field, but shows unusual talent at an early age, this effect must be greatly magnified, and I can sympathize with the wish to avoid it.
The difference, though, is that people within the field really don't behave like that. Their attention really will be on the work itself. There may be just a little bit of extra amazement at the author's age (since physicists are human too), but it won't overshadow the work. It seems that you've tried to reassure the student of this, but he's not convinced.
So perhaps you should try to put the student in touch with one or more people in your field who has had the experience of being a "prodigy", and gone on to be a successful, mature researcher. This person should hopefully be able to better relate to your student's concerns, share their experience of what it is actually like to enter the field from such a background, and offer guidance for managing unwanted attention or similar issues. This might help the student understand that his fears are unfounded, and encourage him to go ahead with publishing under his own name, which I think we can agree is ultimately what's best.