It's difficult to give a firm answer without seeing the contract between your friend and his client. However, in general, I think it unlikely that the client would be able to claim damages.
Disclosing the client's name in this way is definitely unprofessional. Your friend made a serious mistake in doing so (although he did the right thing by recitfying the mistake straight away). But that doesn't mean that he was in breach of contract. Unless there was a specific term in the contract covering this point, I can't see how the client can do anything about it.
Even if the point was covered in the contract, the client would have to show that he suffered some damage or loss as a result of the disclosure. I don't know the circumstances, but it's hard to imagine how a client would suffer loss or damage just because a few people know that a particular graphic designer is working for him.
A more important point is that the affair will likely lead to a breakdown in trust between the client and designer. Your friend needs to go out of his way to restore good relations with the client, if only so that he can complete the work he is contracted to do, and get paid for it.
At least, that's my layman's opinion. If it looks likely that the client will take legal action, your friend will of course need to get proper legal advice.