A determiner denotes whether the noun is referring to a definite or indefinite element of a class, to a closer or more distant element, to an element belonging to a specified person or thing, to a particular number or quantity, etc. As a determiner, we use ‘some’ before nouns to refer to an indefinite quantity. It is generally used in sentences with a positive tone. So our sentence becomes – Would you like to have some coffee? my host asked.
Hence option A is the answer.