I guess it's a port related issue. The incoming connection (from outside – internet) is blocked by your mom's modem/router. It's a normal modem/router behavior.
Screen sharing usually works through port 5900 as stated on this Apple's page: TCP and UDP port reference in Remote Desktop - Apple Support
Locally (when you was at her home) it worked because the connection was within the local network – not through internet.
In order to get it working through internet you need to forward the mentioned 5900 port on your mom's modem/router. Moreover you need to know your mom's external IP address to reach her remotely.
Considering she is not familiar with everything I wrote above I suggest another route.
Sharing the screen via a FaceTime call is a good option assuming that both of you use Apple devices. Follow this Apple's page to discover more: Use SharePlay to share screens in FaceTime on Mac - Apple Support (IE)
Basically, it doesn't need any particular action or setup from the involved users because an Apple server between them is managing the connection.
Other solutions involve third party services/apps which work in a similar way.