First, I'll assume that the student did actually ask a very silly question.
One thing that needs to be always taken into consideration in situations such as this is: Is the student aware of the silliness of the question?
While it may be obvious to you that the question is indeed silly, it may not be obvious to the student. You haven't posted any in-depth information about the whole background of the case, but it could very well be that the student is misunderstanding something extremely fundamental. You claim that the question is not related to the lecture, but at the same time, you say that you have in fact previously talked about the issue. In student's mind, that could somehow relate the issue to the lecture.
One more thing that could be very problematic is if student has a flawed thought process. This can cause the student to relate things that are not actually related and to find cause and effect relationships where there are none.
For example, due to administrative problems at my university, we were required to take an advanced course in one field before taking an introductory course. I already had some experience in that field, so a number of my fellow students asked me questions when they had problems with that exam. Some of them were actually very silly! Often, the silliest were for me the hardest to explain since that required figuratively digging through their brain in order to find the cause of the reasoning that got them to ask me the question in the first place. It often turned out that there was a mistake in reasoning somewhere or that they misunderstood something at a very basic level. One more thing that I noticed is that some people would try to avoid fully understanding the issue. Often, after explaining A for example, I'd have conversation going something like "Is A clear?" "Yes!" "Really?" "Yes, really." "100% clear?" "Yes, crystal clear!" and then it turns out that it wasn't clear.
Just explaining the initial problem and stopping there in cases such as the one I mentioned in the previous paragraph is just treating the symptoms of a disease and not the cause itself.
Another thing I'd like to mention is that (at least in my environment) those who ask questions do actually care about the subject they're learning and in general have a valid problem. It's usually those who don't ask questions that have no idea what's going on.
Next, I'll write a bit about "You're paid" problem itself.
It's commonly repeated that there are no bad questions, only bad answers. In my opinion, whoever produced that piece of wisdom didn't see enough questions.
You need to make a policy explaining what you should do in cases of inappropriate questions. Are you or are you not payed (or for some other reason expected) to answer such questions? What should you do in case you get a student who's too stupid to pass the course you're TA-ing for? What about students who can't form an answerable question in their mind? What about students who ask malicious questions? I was quite surprised to hear from one student at my school that he asks a certain TA senseless questions just because he like to see her struggle to answer them.
Only thing I can advise here is to think hard about what you are and are not expected to do. Talk to your colleagues about that and talk to your superiors about that. Make the limits clear to yourself and to your students. This way, when you come to a similar situation again and you explore all other options, you can honestly and with clear consciousness respond to "You are paid to answer my questions!" with "No, I'm not!"
Finally, I'd like to recommend to you to keep the tension low if at all possible. In this particular case, you mentioned that the student insisted that you answer the question many times. That probably means that the question is important to him. Is it normal that students should ask you more than once to answer a question? Is that expected in your culture? I never had to ask a professor or a TA question more than once in during my whole education so far. You provided an answer that upset the student. Were you aware that the student would be upset by such answer? If you were, think about why you gave that answer. Was that student confrontational before? Does he ask too many questions? Were you having a bad day? Did you give such an answer just because the student asked at an inappropriate time?