Suraya vs Suryaya
I have been doing a lot of thinking about this and I'm now a firm believer that these are two different terms that are ultimately derived from the same root (Assyrian). But historically I can at least attest that Eastern Assyrians do not use the term "Suryaya" to describe themselves, it's always "Suraya". Can't say much about Western Assyrians other than the term "Suryoyo" is what's being used more these days, although a few Western Assyrians have told me that in the homeland and among the older generations, it was "Suroyo", and the term "Suryoyo" was pushed more by the church in recent times.
Anyways, let me discuss why I think the two terms are different and why we need to drop "Suryaya/Suryoyo" and just stick with the natural "Suraya/Suroyo" as our natural endonym. It is true that in our church history and the early Syriac Church fathers write "Suryaya", not "Suraya", but this term simply means someone from "Surya", because Syria was the region and adding the "ya" to it makes it "Syrian". Even if this term comes from Assyria, Syriac Christianity in itself was born in Antioch when it was under Roman rule, and Antioch was in the Roman province of Syria. This is why you see the early church fathers identify with this term, because places like Antioch and Edessa were basically looked at as "Syrian" cities, and these church fathers that wrote our early church history identified with this region.
On the other hand if we move east to where our people come from, the term used is "Suraya", not "Suryaya". Oddly enough modern scholarship thinks that Suraya is just a short form of Suryaya, meaning overtime we just lost the usage of that yod. There's a problem with this. Unlike Western Assyrian where it's mostly contained in one place (Tur Abdin). Eastern Assyrian is spread out across a lot of different geographical regions and there are so many diverse accents. We're talking Nineveh Plains, Nohadra, Zakho, Erbil, Hakkari (Which in itself has many accents), Urmia, Salmas, Bhotan, Siirt, Cizre, and others. They all say "Suraya", not "Suryaya". If we dropped that extra yod from the word you'd still see a few accents using "Suryaya", it would be some kind of crazy coincidence that all these diverse accents that are spread across all these different areas decide to change from Suryaya to Suraya. The other thing is our Jewish neighbors that also spoke Aramaic called us Suraye, not Suryaye.
So comparing to the etymology of Suryaya (Someone from Surya), Suraya would be someone from "Sur". Given our geographical location and how close "Sur" is to "Assur" in sound, seems self explanatory to me.