My friend said to me one day: "We have a mice problem at UNI". Is "a mice problem" grammatical as opposed to "a mouse problem"?
The word "mouse/mice" in the phrase "mouse/mice problem" is an attributive noun (also known as a noun adjunct). Generally, in the English language, attributive nouns take the singular (as in marriage certificate and bottle opener—see this dedicated article by Nordquist for more examples). Nevertheless, the above link about attributive nouns also cites a study stating that "[i]t is normal that the first or attributive noun of a sequence will be singular. Yet studies of recent English . . . have noted the apparently increasing variety of formations with a plural attributive noun [emphasis mine]."
One could argue that some attributive nouns are apt to take a plural form; "a lice problem" might sound more natural than "a louse problem", especially because an infestation issue implies numerous vermin. However, the wording "a mice" at the beginning of "a mice problem" can certainly sound awkward, even though "mice" is being used as a modifier. My best recommendation would be to use whichever phrasing sounds best in the particular context, or better yet, reconstruct the phrase to read "a problem with mice".