It has to be SHE OFFERED or SHE OFFERS. SHE OFFER is incorrect. Then the answer could be B or C, though B is more likely.

We would say: "She offered me a cup of tea," if we want to express that she is giving you one cup of tea. You're right that C is technically correct, but we don't use that phrase often, or for that purpose. You could say "She offered me only one cup of tea," if you want to emphasize that she offered you just one, not more. You can say "She offered me another cup of tea," when she has already given you one or more, and she is now offering you some more tea.

B. She offered me another cup of coffee

I think 🤔 , this answer is ( one cup of tea).

"She offered me ___ cup of tea." would be the correct sentence structure. "offer" is providing someone with a proposal or discount or a business offer "what's the offer?". "offered" is giving something. "She offered her help." "She offered to be my friend." The answer to your original question could be B or C. "B" if you already had a cup of tea and "C" if she only gave you one cup, but in America we would use "a cup of tea" to show a single cup.