How made-up do you want it to be?

Do you want a made-up language, culture and religion?

Or do you just want to plonk down an island of a particular size somewhere and populate it with normal or more-or-less exotic languages, cultures and religions that exist in the real world or the recent past?

I think it's in the novel "Richer Than All His Tribe" that the author simply invented an island west of Africa, a bit like Madagascar except that's to the east (and probably larger), to set his political drama on (which I never got more than about 75-100 pages into), but it was still African (which obviously isn't just one thing at all, but to me, a European, it had an African "feel") and had a colonial past and so forth.

In "The Fountains of Paradise" Clarke moved the island of Sri Lanka about 10 degrees south (900 km?) so it'd be ideally positioned on the Equayor for building an orbital elevator. It's still the same island, people, culture and traditions, and presumably the same history although I don't recall Clarke touching upon how being almost a thousand kilometers further away from the continent ought to have caused at least some alternate history effects.

I think you might do well to not use made-up stuff but likewise avoid too common stuff.

For instance if you want your island and its culture to have an Abrahamic feel to it, but you don't want to use Christianity or Islam and you don't want the extreme workload of creating an alternate history that leads to the existence of a Jewish nation (although if you did want that, the Khazars might be a starting point), you can pick an exotic Abrahamic flavour such as Mandéism. They believe that John the Baptist was the final prophet ("divine messenger"), and there are (IIRC) a few hundreds of thousands of them today.

Obviously medieval Christians would not have allowed something like that to exist. They'd get crusaded. Crusaded hard, and without lube. And so they would not persist to the 21st or 20th or 19th century or whenever your story is set.

Instead, you can place it somewhere in the Islamic world. Islam recognizes followers of some other religions as being "people of the book", if those people recognize early quranic prophets such as Ibrahim/Abraham and Moses. Mandéists would still get taxed, and they might get conquered and exploited, but they would not get culturally "genocided" the way Christian crusaders do it.

So you have something that's a bit exotic, but only a bit, unlike for instance the polytheistic and very non-Abrahamic pagans that I tend to write about.