My friend asked me today: "What fruit would you like to eat?" and I pondered which phrase will be correct in such situation. "I don't think I want a fruit now, but thank you for the suggestion / invitation". Is these words (suggestion / invitation) are correct in this context?
As your friend's question implies they will give you a piece of fruit, "suggestion" does not fit very well. "Suggestion" would be appropriate if your friend gave you advice on a course of action, but not if they are offering you something tangible. On the other hand, "invitation" suggests something more than a piece of fruit - if your friend offered to buy you a meal, it would be more appropriate.
As a native BE speaker I would almost certainly respond with something including "Thank you for the offer" or "for offering".
Of the two, I would favor "invitation", as that seems more of something to thank another person for. However, in ordinary informal conversation, I would be far more likely to say something like
Thanks, but I don't think I want a fruit now.
Not just now, but thanks anyway.
Thanks, not just now.
The last of these is more perfunctory, and in my view therefore less polite. I would use it with family or a good friend, but not an acquaintance I was trying to make an impression on, nor a date (if I were still dating).
I don't think I want a fruit now, but thank you for the invitation.
seems over elaborate. For the matter of that, i would be far more likely to phrase the question more simply as:
Would you like a fruit?
with a gesture to a fruit bowl or display, if one was in sight.