Make sure you can find the remote computer on the net, and that requests get from the router to the computer.
Most ISPs give dynamic IP addresses, not fixed ones. That means that the set of numbers that uniquely identifies the modem on the internet can change. For example, one day it could be 192.0.2.0. The next time you want to connect it could be 192.0.2.20. To make certain that I could always contact mum's computer, I used the services of a company called DynDns. In a nutshell, I registered a name for mum's computer with Dyn - let's call it kevsmum.example.com. Then I configured mum's router to tell Dyn the IP address that corresponds to kevsmum.example.com. The router updates the IP address periodically.
Whenever I want to contact mum's computer, I use the host name kevsmum.example.com, rather than the numeric IP address, and Dyn route the connection for me.
Many routers have a setting to specify the company and host name details; Dyn is not the only company to do this. If your router doesn't (for example, the Virgin Media superhub doesn't), you can run tools on the computer that do the same thing. Again, the Dyn web site has details.
[Update 2013-07-09] I'm in the process of moving from DynDns to Free DNS.
Finally, I allocated a fixed internal IP address to mum's computer, and configured the router to route SSH requests to this IP address.
Make sure you do this before you leave - especially if you live at a distance.
You need to be connected to the internet - but not on the same network as the computer of the person that you're caring for. I took along a laptop with a USB broadband dongle. This meant that I could access the net over a 3G connection. The downside of it was that I could only get a signal on the dongle outside. So I kept moving between the router (indoors), and the dustbin|trashcan (outdoor stand for the laptop). Sheesh!
I needed an easier way. In summer the dustbin pongs; in winter it's cold! My Android phone gets a good 3G signal indoors, but my contract forbids tethering (using the phone as a modem for another computer). Fortunately, Google Play has several good, free, SSH clients. I installed ConnectBot. If you can successfully establish an SSH connection, you're good.