The problem is that your sentences are mainly either imperatives or statements. The imperatives are unsuitable for a test, and they would come over as too direct and abrupt. The instructions "Give me a second" and "Give me time" and "Hold on, I'm thinking" are too direct for this context. You could say these to friends and family, but you are not in a position to give orders to the examiner. They sound rude.
The only one of your imperatives which is polite and friendly enough is the first, as it sounds less direct: "Please let me think about that for a while" is OK. Note that I've changed 'it' to 'that', to show that you're referring to something that the other person has just said. 'It' isn't wrong, but 'that' is better.
"I'm so sorry but I want some time to think about it" is not at all natural. "I'm so sorry" is too much of an apology ( there's no need to sound so contrite) while "I want some time" sounds rude (see above).
"Let me think" is informal, but it might be OK if you feel the atmosphere between you and the examiner is relaxed enough.
So... what's the solution, if neither imperatives nor statements are suitable? Easy - go with QUESTIONS. Your only question ( lacking a question mark!) is fine:
"May I have some time to think about that?"
If you want to sound more friendly and relaxed ( less formal, but still polite), you could make up some questions using this as a formula:
2. .... you (just) give me/let me have OR I (just) have
3. .... (just) ...a minute/ a moment ( or two)/ a few moments/ a little time/ some time
4. .... to think about that, please?
For example, "Can I just have a minute to think about that, please?" or "Could you give me a moment or two to think about that, please?". Any questions from the above examples will be fine and appropriate for the context - polite, friendly and natural.