In the phrase "a button click", what are the parts of speech of each word?

"Button click" is a compound noun. Each can also be nouns on their own. Click can be a verb, such as in "Next you will click the button." I suppose one could debate whether the compound noun "button click" is made up of a noun and noun or a noun and verb, but I would say it is a noun and noun but that the distinction is irrelevant. When I perform a click [noun] on "Post", this answer will appear. As when I click [verb] on "Post".

Compound Nouns A compound noun is a noun that is made with two or more words. A compound noun is usually [noun + noun] or [adjective + noun], but there are other combinations (see below). It is important to understand and recognize compound nouns. Each compound noun acts as a single unit and can be modified by adjectives and other nouns.

From: https://www.englishclub.com/grammar/nouns-compound.htm

EDIT: Reread your specific question. If button were not there, click would definitely be a noun. It is the object of perform.

The word group

a button click

is a noun phrase (NP) whose head is the noun click. One way to see that click is a noun is to realize that it can be pluralized: two button clicks. Incidentally, this also shows that it is click that is the head of the NP (rather than button, which remains in the singular).

The word button is also a noun. It is clearly not a verb, or a preposition, of a determinative, or an adverb. The only thing it could plausibly be, apart from a noun, is an adjective. But it is not an adjective either, because it can be modified by an adjective: a large-button click. If it were an adjective, its modifier would have to be an adverb, so largely.

Finally, a is an article, which older sources classify as an adjective, while contemporary grammars would say it is a separate part of speech, a determinative. Its grammatical role in the NP is that of a determiner.

We say that button plays the role of a pre-head internal dependent in NP structure.

In general, internal dependents can be either modifiers or complements. The most important characteristic that distinguishes the two is that a complement must be licensed by the head noun (shortly it will become clear what this means). CGEL says (p. 439-443) that with pre-head complements, we

typically have close paraphrases involving post-head complements where a forced choice of preposition is observable. Compare, for example, an alcohol ban and a ban on alcohol, a wrist injury and an injury to the wrist. Similarly, corresponding to a legal adviser ([iii]) we have an adviser on legal matters.

It is the fact that we don't have a choice which proposition to use that makes it licensed (in the sense that other prepositions then aren't licensed).

In our case we would have a click of a button, where of cannot be replaced by any other preposition. Thus, it is licensed; thus, button is a complement of click.

Compound noun vs. modifier+head pair

A priori, there is a possibility that button click is a compound noun. The way to distinguish compound nouns from mere modifer+head pairs ('composite nominals', as CGEL calls them) is that in the latter case, the two nouns can enter separately into relations of coordination and modification (CGEL, p. 448-451). As an example, consider London colleges. Is it a compund noun, or a modifier+head pair? The acceptability of the following shows that it is the latter: various [London and Oxford] colleges; various London [schools and colleges]. In contrast, ice cream is a compound noun, because we can't have ^*[ice and custard] creams; we have to say ice creams and custard creams. Similarly, we can't say ^*crushed ice cream when we mean cream of crushed ice. In contrast, we may say London theological colleges, where colleges is separately modified by theological. There are other criteria that can be considered as well.

As for a button click, I believe the following shows that it is not a composite noun:

a left- button click (examples)

An additional consideration is this: compound nouns (like bus stop) tend to have separate entries in dictionaries (like this), whereas button click doesn't have one in any of the dictionaries I looked at, including the OED. Here they are: Merriam-Webster, Lexico (formerly Oxford), American Heritage, Collins, vocabulary.com, Macmillan, Cambridge, Wiktionary, Webster's New World College Dictionary, Fifth Ed. (via yourdictionary.com), Infoplease, dictionary.com, and freedictionary.org.