The correct option is D a
The noun in the given statement, 'name' is described by the adjective 'French' here, and it is not specified which name in particular, hence the indefinite article 'a' is the correct option, since the noun and adjective both begin with a consonant, and not a vowel, hence option A is incorrect. 'The' is incorrect because the adjoining noun to the adjective is not specified. 'French' in the first part of the sentence is an adjective since it describes the noun, whereas, in the second part, it refers to the nationality. No article is used in the second part, but the option D is incorrect for the first part as we not referring to French names in the general sense.