I'm not talking about things like hardware detection or device support -- two things I hear a lot of -- because those already have legions of folks behind them and aren't specific to any distribution's approach. Rather, I'm thinking about things which a distribution can improve on its own to the point where it can serve as a positive example for others to follow.
Here's a starting list of five, in no particular order:
1. Improve documentation. This is something I've written about before, and I got some wonderfully positive feedback re: Ubuntu about that subject. The community surrounding most any Linux distro is typically quite supportive -- but there are going to be times when you need canned answers at your elbow, too.
2. Provide specific and ongoing support for one or at most two applications to accomplish a particular job. For instance, if I have five applications that provide backup/restore functionality in various forms, it would be nice to have the most robust, fully featured backup tool (or, again, at most two of them) supported directly by the community. In this case, "support" means active development, either in the form of a community-branded fork of the project, or contributions back to the original codebase for the program. The number of applications they support in this fashion can vary depending on the distribution in question and could even revolve periodically.
3. Make it an ongoing project to reduce the amount of command-line hackery that needs to be done to accomplish relatively simple things. This doesn't mean "do away with the command line"; I'd be nuts to suggest that to anyone. It does mean that a given distro should always provide the user with friendly ways to accomplish common things -- like editing supported resolutions for a video monitor -- that often depend entirely too heavily on editing configuration files. (There's always going to be a need to drop to a CLI and edit directly no matter what the environment, but I say the less someone has to do that for relatively uncomplicated operations, the better.) Each distribution covers slightly different territory, so the types of common tasks that can be tackled this way would also vary.
4. Create a consistent format for configuration files. Perhaps impossible to ask for, but I'll do it anyway. All configuration files, especially when it comes to config files for individual applications vs. config files for the OS, seem to be wholly inconsistent with one another in terms of format and presentation; let's clean them up. Granted, it's one of those very long-term projects which would essentially involve rewriting a lot of things -- I have no delusions about how difficult or involved this would be -- but in the long run it would make things far less obscurantist and more self-documenting. Some programs already do this -- I think the JWM window manager uses XML as its config-file format, which makes it very easy to work with -- but it would be nice to see this conceit adopted as globally as possible.
5. Provide a live distribution. The majority of distributions I've run into provide live distributions -- a CD image you can just burn to disc, boot, and use without installing anything, or write to a USB flash drive and boot with if your system supports that. There's no better way to get someone hooked on a given breed of Linux than with a live distro. Me included.