Renaming the current file in Vim

The command is called :saveas, but unfortunately it will not delete your old file, you'll have to do that manually. see :help saveas for more info.

EDIT:

Most vim installations have an integrated file explorer, which you can use for such operations. Try :Explore in command mode (I would actually map that to a function key, it's very handy). You can rename files with R or delete them with D, for example. But pressing <F1> in the explorer will give you a better overview.

If you use git and already have the tpope's plugin fugitive.vim then simply:

:Gmove newname

This will:

1. Rename your file on disk.
2. Rename the file in git repo.
3. Reload the file into the current buffer.
4. Preserve undo history.

If your file was not yet added to a git repo then first add it:

:Gwrite