I don't have any type of historical credibility like some of amazing experts here, however I am a plains Indian and while my vocabulary isn't the greatest I know a few words. Mustus or moostoos depending on dialect or preference for English spelling is plains Cree (tribal plains Indians inhabiting lands North of the Lakota, Blackfoot (Sioux peoples ) and historical rival of the Sioux (northern Alberta, Saskatchewan, Manitoba and Ontario, extending into northern Quebec)

Also having travelled a bit and picking up the occasional Sioux word I know Taunka is the Sioux word for Buffalo.

I'm curious what sort of argument you were having, so if you feel like adding some info on that, I'd be interested to hear it.

Anyhow, to add to what others have said:

For Lakota: ptéȟčaka is the general term. tȟatȟáŋka is a male buffalo, pté is a female buffalo. You can you use this dictionary to look up related terms. This dictionary has quite a few buffalo related terms, but uses a slight older, more simplified orthography.

In Omaha-Ponca, a male buffalo is tenuga and a female is temi^n ga (no space, but I'm not sure how to get Reddit to only superscript only one letter at a time. This link will take to you the start of the list of buffalo-related terms as well.

In Pawnee, a male buffalo is kiwiiku' and a female (as well as the species in general) is taraha'. Another dictionary for you.

Off hand, those are the ones I knew would have easy access online for you. If you want some more examples, I might be able to dig up some more for you once I get home tomorrow.