I really like the name LitecoinFree (describes NYC specs very well). You don't? But I am not planning to re-name the existing chain anything. Or market it. Or try to create extra value from it. Just ensure it is healthy and confirming transactions every 30 seconds free of charge. And give the new chain enough time to be real-world tested and proven to be worthy of replacing our amazing (slightly outdated) existing codebase.
I'm fine with the name, Charlie Lee might have a different opinion.
The issue that concerns me and should concern you is that users will be confused about which chain they are on and will experience the same issues that they recently experienced with the chain split. This was only a chain split, when the hard-fork happens they won't be able to recover from attempts to send transactions across chains. If you rename your chain to "Litecoin Free" much of this confusion could be avoided.
The NYC dev team has said nothing about implementing replay protection. Have you looked into or thought about the implications?