While none of them have gone very far, there have been suggestions to divide New York into multiple states, usually NYC and Upstate. However, I've seen few serious proposals for what the name of the new state(s) would be. This can be for a NYC/Upstate divide, Western NY/NYC+Hudson River Valley, or any other partition.
It would make the most sense for NYC to keep the title of New York, but Mayor Fernando Wood proposed the name Tri-Insula when he suggested secession in 1861. Hamilton (after Alexander Hamilton, who made his career in the city) could also work. Upstate New York could use the name of one of its prominent cities - Albany and Buffallo - or it could use a geographic name like Hudson or Niagara (the later of which was has been used as a proposed name for a state consisting of western New York).
Putting this in post-1900 since that's when partition and secession became popular, but I'll accept pre-1900 PoDs.