Syncthing if you want continuous sync. Rclone if you want periodic rsync-style copies. Both have web UIs, although they're definitely aimed at the power user and not novices. If they're non-technical users you probably want to setup a nextcloud or similar system so they get a more dropbox-like experience.
edit: reddit.com/r/selfhosted always has good threads on self-hosted web file managers too
To clarify, priority is browsing, copying and moving stuff on same remote server, not copying data between machines. Thanks, will try these anyway. Trying this too https://cloudcmd.io
Edit: Cloud Commander works, gives you two-pane webgui with a row of Fx buttons below. F5 copies files locally.
"lsyncd" is really good for efficient, continuous sync. If you have password-less, public-key SSH auth, lsyncd is a bless (using it for millions of files and folders and doing good)
edit: reddit.com/r/selfhosted always has good threads on self-hosted web file managers too