What does illini or illiniwek mean in native American language?

The specific language is called Miami-Illinois, spoken by many small tribes living in the area of the upper Mississippi river valley (Peoria, Piankashaw, Miami, Wea, Michigamea, Cahokia, Kaskaskia and many others).

In that language the Word for "man" or "person" is alenia. Despite the similarity of this with Illini or Illinois, it is accepted that both illini and illiniwek are white European (French) versions or approximations that come from the term irenweewa or ilenweewa, meaning "he speaks the Illinois language" (effectively "he is one of us"). The plural of this would be irenweewaki ("they speak our language")

But the self-designation for these people was normally inoka.

So the answer is that illini and illiniwek mean absolutely nothing in any language, but they are corrupted forms of the Words mentioned above. This is typical of most of the names mis-applied to native tribes by explorers unfamiliar with their languages.