Y'il, I guess is how I would spell it?
At first, we called her "little kitty" because we couldn't think of a name. Then "Littles" kind of stuck, because even when she was fully grown, she was very small, especially in comparison to our other giant cat. From "Littles" it went to "Li'l", and from "Lil" it went to "Yeel" or "Yi'l", because we all like to talk in baby voices to our animals, and we would often call her a "yittle baby."
When we were talking to each other about the cat, like "go feed her some water", we would usually say "Yittles".
Often times, when talking to the cat, it wasn't just one Y'il, but two, often meshed into one. "Yiyi'l" I guess? It's too hard to try to explain through text...