Eugene (do they call you Gene?) Volokh writes:

While we're at it, why are there gender-generic terms "parents," "siblings," and "children" -- and, in English, only a gender-generic "cousin" -- but not "auncles" or "niecews"? (No, I'm not serious about those two particular suggestions; I'm noting the absence of any such word.)

After doing a minor amount of research about why there isn't a different word for M/F cousins, I suspect the answer is because when the word came over from Latin (via old French), we lost the endings that differentiated M from F and end up with the same word for different genders.

The Latin root words are consobrina (F)/consobrinus (M), meaning cousin on the mother's side. Note that there's a different set of words for cousin on Dad's side (sobrina/sobrinus) and another set of words for cousins that are from your Dad's sister or your Mom's brother (amitina/amitinus).

Disclaimer: I don't pretend to speak Latin, but I am pretty handy with Bartleby's.