First of all, "in particular" needs to be set apart with commas. What, in particular, does a person gain when he or she becomes a serious collector? That clarifies (for me at least) what it modifies. It modifies, "what". The difference between: What, in particular, does a person gain when he or she becomes a serious collector? What does a person gain when he or she becomes a serious collector? Is that the speaker is asking for more detail. I think "in particular" is an adverbial phrase.

"In particular" is a prepositional phrase ("PP"). Such phrases most often function adjectivally, but they can act adverbially, too. Before getting to your example, let's take a look at some examples which may be a bit more clear-cut, but still using the interrogative pronoun "what" as the object of the sentence: 1. "What in the collection did you enjoy?" 2. "What, in your opinion, did they enjoy?" Now in 1, the PP modifies the pronoun "what". Conventionally, we say modifiers of nouns and pronouns are called adjectives, so we can say the phrase acts adjectivally. Note that you could move the PP to after "enjoy", and the sentence would mean the same. Conventionally, we call modifiers that don't attach to nouns and pronouns adverbs, whether they actually modify verbs or something else. (I'm not sure it's a very useful distinction, but that's the traditional view.) In 2, the PP is actually modifying the whole clause; it is a "sentence adverb". Notice that it is set off by commas, and the PP could be moved to the beginning or end of the sentence with the same effect. So having said all that, what is your example? It could probably be interpreted either way. But without the commas, I am more inclined to look at the PP as adjectivally modifying "what". Another way to get a handle on it is to think of a declarative statement as an answer to this question, using a true noun, e.g. "One gains wisdom in particular".