Side projects do influence the hiring process positively. The reason is that they show the developer in question does what they do because they enjoy it rather than do it just to get paid. If it doesn't show enthusiasm for work it at least shows drive and ability to contribute.
And in regards to how to choose a project and how each influences hiring managers; there are two routes to go down here. If you have the time you can even do both.
Universities/colleges tend not prepare you for larger scale projects and things like version control so a great way to develop personally as a developer is to involve yourself in some open source project, either through SourceForge or GitHub for example.
This will do two things, it will teach you what most people learn on their first months in a job straight out of study, that's the two are nothing alike. As well as that it will show an employer that you can
- work in a team as a developer & collaborate
- deal with large scale projects (in terms of code base and contributors)
- it will also go a bit further in showing that have capable development skills as they will be able to view your code independent of the interview
Smaller personal projects are useful in showing you have drive and design skills. More importantly they will help you develop better design skills. If you plan to bring this code along to an interview bring design docs (UML for example).
Also for completeness of answer, for non web developers; if you want to improve your skills as a developer with personal projects attempt a simple compiler, there's mountains to be learned there about not only development but also your platform or if you are very adventurous a very simple OS. Both of these projects never need to be completed to learn a great deal from.
Do smaller exercises all the time, so as not to make the mistake many graduates do, if you can't think of any ideas Project Euler has plenty.