What should I do if I don't know the answer to a question posed by a student?

I agree that honesty is the best policy, and it's too bad if you're in a situation in which you feel worried about admitting you don't know the answer. You shouldn't try to bluff, by pretending you know but don't have time to explain or by giving an intentionally vague answer. However, there are ways of handling it more smoothly than just saying "I don't know" and leaving it at that. Depending on the circumstances, you can say "That's a really interesting question. I haven't thought about it, so I'll have to look into it, but let's talk about it in office hours." (Or you can promise to return to the topic in the next class meeting if it's really relevant to the course and everyone in the class will want to know the answer.) Or "These issues can be complicated. I don't know the details off the top of my head, but the place I'd look them up is Reference Work X. I'd be happy to show you where to find it after class." Or "That's a good question, but it's somewhat beyond the scope of this class. I'd be happy to investigate it with you outside of class."

The key is to respect the student's desire to learn. If you avoid the question or give an answer you know is inadequate, then you're being deliberately unhelpful. If you just give up and admit defeat, then at least you're being honest, but the student still isn't finding out what he/she wanted to know. If you respond by pointing the student on the road to an answer, even if you can't supply it off the top of your head, then you've done everything that can be expected of you.

I'm a professor, so I am expected to know the answers, but sometimes I don't. This often involves some minor detail in a programming language.

So I usually say, "That's a good question. I don't want to give a wrong answer, so let me think about that and get back to you." We use a course management system which includes a discussion board, so I will usually then add, "I don't want to forget, so post that on the forum. That way everyone will see the answer." Then in the posted answer I try to explain how I found the information. I find this works well.