Szia!
As for “-gil(ler)”;
Pétergil(ler) ile tatile çıkacağız– Péterékkel fogunk nyaralni.
So it can be said to be the equivalent of Hungarian “-ék”, which is attached to (sur)names. It is used either as “-gil” or “-giller”, no difference in meaning.
It is mainly used in spoken, everday language and the plural ending “-lAr” has been recently used more and/or in formal language instead . Besides, I hear it less compared to the old days.
Another example from my childhood;
Annemgil akşam size oturmaya gelecek - Az anyukám (és az apukám vagy ha valaki más is van, akkor ő is) vendégbe mennek este hozzátok (remélem, hogy jól írtam).
So we can say that it is used with personal names and kinship terms to denote family (or possibly a group of friends) but the plural ending has more or less replaced it. Accordingly, a small frequency list of the possibilities can be;
Kayalar (Kayáék (lit. Kayák))
Kaya ailesi (A Kaya család)
Kayagil(ler) (Kayáék)
Its origin is still debated as far as I know but don't think that it is of Arabic.