I don't use LyX, however I guess it's a backup file. Some editors add a tilde to the names of backup files. Open it with a text editor or any program to read it, look if it's similar to the original .lyx file.
As Stefan said, it is a backup file indeed. If you rename your main .lyx file, a new .lyx~ file will be created, with the new name, while the old one, with the old name, will not be deleted. It is good to have these files in case of a disaster (e.g., power-cut might destroy your document completely or mess it up badly).
There are two things you might want to know regarding backup files: 1. You can tell LyX how often it should create/update backup files, or even disable them completely (which is not a good idea): Tools > Preferences > Look & Feel > User Interface (first two checkboxes under "Document") 2. You can create a directory in which backup files of all your projects will be stored, instead of the current directory: Tools > Preferences > Paths > Backup directory.
