This often proves rather difficult, at least for myself. I'll start off with an example. When I first started to become interested in Linux, a friend decided to lend me a disc. It happened to be Ubuntu 5.10. To this day, I either use Debian, Ubuntu, or Mint. This is because it was my first impressions to Linux, and it's what I accustomed myself to. So this is where you need to decide what to show them.
For instance, decide how competant they are when it comes to computers. While this may be a harsh statement it's very true. If this person is only able to log in and surf the net, don't suggest something such as Arch or Gentoo, rather suggest something such as Mint. If they enjoy a little bit of monkeying around with things, give them Ubuntu, or perhaps even openSUSE. It depends on the type of person
Now what exactly do you show a person to demonstrate Linux. First you have to think about what Linux really is. At the very basic level, it's an OS. It provides a layer to execute programs that people have written, and for some people that's good enough. For others, and in reality the majority of the young population, you're going to have to convince them that Linux is better than Windows.
One of the first things that you should show them is the office suites available. I personally have no use for an office suite, but apparently people are willing to pay 300+ dollars for that software. The next would be photography, such as the Gimp. Show them how to do things that you could do in Photoshop in the Gimp. Show them how the media players work, how to create movies, with something such as Kdenlive. This is what they want to see. Then show them them some of the fancier things. Do a little bit of management through the CLI. How to list files, move, copy, install applications, start applications, configure the os. This will show them that you both know what you're doing and what is available to them on Linux. The last thing you need to do is ask if they have any apps that they couldn't do without, and then show them a suitable replacement for them, and if you can't even find one, then use the opportunity to demonstrate Wine to them.
All this being said, it's a good idea to watch what you show them. Just like the old saying--first impressions are the most important.