Use an Apple MobileMe subscription. Register both your Mom's and your Macs with your MobileMe subscription. Now use the MobileMe Back-to-My-Mac feature. This is a secure connection.
Alternate, setup would be to use Hamachi (VPN)
You could use TeamViewer.com which setup a secure remove screen sharing connection.
You could use LogMeIn.com which also setup up a secure remote screen sharing session.
You could use iChat Screen Sharing. You can get free AOL Instant Messenger accounts for both you and your Mom, then take your Mom through configuring iChat with the account. Now you can use iChat Screen Sharing.
I don't want to use Log Me In Hamchi ($200/year) nor MobileMe ($129/year). I want to use Apple Remote Desktop which I already own. How would I go about using that?
Both Hamachi, LogMeIn, and TeamViewer have free for personal use options.
Hamachi would be the easiest free for personal use VPN solution.
A more labor intensive approach would be ssh tunnels. This involves modifying your Mom's home router to Port Forward port 22 to her Mac, getting a free dynamic DNS name so you can find your Mom's router, enabling Remote Login, Screen Sharing, and File Sharing on your Mom's Mac, then you establish an ssh connection with tunnels for Screen Sharing and File Sharing from your Mac to your Mom's, then you can use ARD to securely connect to your Mom's Mac.
I would suggest checking out Slink from Slinkware (also available on the Mac App Store). It's easy to set up, there's no monthly fees, and it works with most Bonjour services on the remote network. I've been testing it out for telecommuting for the past few weeks and it's been working great.
The only drawback is that Slink has only limited support for ARD -- it only supports the screen sharing feature, so there's almost no benefit to using ARD instead of regular Screen Sharing when using Slink.
However, I just came across this page with relatively easy VPN setup instructions:
I just tried it out and it seems to be working well so far. These instructions are by far the most straightforward and easy to follow instructions for setting up a VPN server in OS X that I've been able to find.