Explanation

In the phrase "Frailty, thy name is woman," the word "thy" serves as a possessive form that indicates ownership. Specifically, "thy" is the archaic or poetic form of the second-person singular possessive adjective “your.” This means that it modifies the noun “name” to show that the name in question belongs to the second person (in this case, it could be interpreted as belonging to women in general or to the person being addressed).

A possessive adjective, unlike a possessive pronoun, is used directly before a noun to indicate to whom the noun belongs. For instance, in modern English, we say "your name" as opposed to using "your" as a standalone possessive pronoun such as "yours."

Thus, within the context of the sentence, "thy" connects clearly to the noun "name," qualifying it and making "thy" a possessive adjective. The options "possessive pronoun," "reflexive pronoun," and "adverb" do not apply here, as they serve different grammatical functions. Therefore, the correct choice is B: possessive adjective.