I just use 'ssh -X' if I really need to run a remote graphical application.
But it is really slow.
The first you need things therefore are 1) an administrator account on the
system and 2) the "ssh" package installed there. Then, assuming that the
system is behind a NAT router (unfortunately the norm), you need to configure
it (through a Web interface) so that 1) the computer is always given the same
private IP address (you will need the MAC address; 'ip a' shows it) and 2)
incoming connections on the SSH port you will use (22 by default) must be
directed to that computer.
If the computer is sometimes connected by Ethernet, sometimes wireless, then
there are two MAC addresses, you will define two permanent private IP
addresses and direct incoming connected on two different ports to those two
private IP addresses. You will then need your mother to tell you how she is
currently connected to the Internet (or simply try the other port if your
first attempt to connect by SSH failed). You will also need her to tell you
what is the (public) address of her router because it is the one you will
connect to and, unfortunately, it often is dynamic. Just put
http://whatismyipaddress.com (or any other similar site) among her Web
browser's bookmarks.
Have you done all that (which, by the way, is not GNU/Linux specific)?
