I am not sure if my experience is directly relevant here, but I am sharing it in case it is. Also, my mum is a silver surfer too, and I know how tricky it can be helping out over the phone....
I sometimes use an external usb wireless adaptor, and this occasionally mis-behaves in the way you describe, in that it fails to make the connection, keeps asking for the password even though it already has it, and generally being irritating.
In my case the solution is very simple, I just unplug the adaptor and plug it in again. Very often when I do this it connects right away with no further fuss. It is as if the adaptor occasionally fails to boot correctly, and re-booting it by un-plugging and re-plugging it fixes things.
In your mother's case, if the adaptor is built-in, the solution may be a bit more work. One way might be to try a re-boot of the computer, but it may be better to try disabling and then re-enabling the wireless from the top-right toolbar first.
Less drastic than re-booting the computer, if there is a hardware switch to toggle the wifi try turning it off and back on.
Another thing you can try (but it might be difficult explaining this to your mum over the phone) is using the
rfkill command to check the wireless from the command line. These are the commands that may be useful:
sudo rfkill list all
This will show the state of all network devices and the names of them.
sudo rfkill unblock <name-of-wifi>
which unblocks the wifi device. In view of my comments above, there is one further variation that may be handy for toggling the wifi:
sudo rfkill block <name-of-wifi>
which will lock the wifi device, disabling it in software. Re-enabling it with the unblock command might just produce the desired effect (but no promises!).