Option 1, the "quick hack": The variable sys-distro in the files info.json in the snapshotdirs contains the system name, so you can do some scripting to change this every time a snapshot is created. (On most systems the snapshots dirs are the ones of the type /run/timeshift/backup/timeshift-btrfs/snapshots/DIFFERENTTIMES/)
Option 2 (untested), the cleanest and easiest solution: Change /etc/lsb_release and/or /etc/os_release. See also option 3 for the background on this.
Option 3 (untested), fix it for the whole world with code: I did some quick digging into the timeshift source code. (With "quick" I mean, some educated guesses were involved...) and this is what i found:
sys-distro is generated in write_control_file() and is based on the passed argument distro_full_name
current_distro.full_name() is usually passed for distro_full_name
full_name() is generated based on vars named dist_id, release and codename
- These vars are all generated by
get_dist_info()
get_dist_info() gets all this info from /etc/lsb-release and/or /etc/os-release (which is why is suggested option 2)
write_control_file() is defined src/Core/Main.vala. It's probably best to only change that last step.
Note: All options are for future backups, to do it for already created backups use option 1 and change the info.json files that already exist