I had been shown this program, Newton in high school. It was programmed by Paul Keet. It has some good, and some bad.
The good:
- easy to use
- allows for the use of satellites
- allows for static focusing on bodies other than the sun
- 2-dimensional, makes for easier use

The bad:
- Programmed for 1993 hardware, so it doesn't make use of faster processors
- inaccurate - adjusting the speed to anything but utterly slow makes it very inaccurate, which goes back to the previous point
- it resets itself every time the user switches away from the window, so it can't simulate in the background. Since it was 1993 programming, that'd mean Windows 3.1 - no true multitasking
- 2-dimensional, makes it more complex

I have tried Newton's Aquarium.
It is more capable, features full 3d motion (which can be good and bad), better graphics, and what appears to be an improved engine. As an added bonus, you can also change the gravitational constant of the Universe.
The bad - camera control. I can't seem to get it to lock onto a planet the way Newton by Paul Keet does. And sometimes the coordinate axes will spontaneously shift. For example, if I manage to set a satellite in motion around a planet, and let's say its velocity is 29,000m/s in the x-direction. At some point, I'll go back to it, and it'll be a combination of vectors in the x and y direction, whose resultant is still 29,000m/s. However, I don't think I should have to perform manual vector addition because the program decided to shift its coordinate system.

Does anyone happen to know of any good solar system programs that I can tinker around with? I'd like to be able to do things like alter mass of bodies, and add new ones to the system.

http://orsa.sourceforge.net/index.html

Open source too, is it? Very nice, I shall have to tinker with this.

I'll have check out the programs you and hendrick mentioned, as I am not familiar with them. They look really interesting. I'm not sure what you are looking for exactly, but it sounds as if Celestia is another open source program that simulates the solar system, and more, and it allows for certain kinds of customization that might be interesting to you.

http://www.orbitsimulator.com/gravity/articles/what.html
This is a physics\gravity simulator that capable of very complex simulations involving hundreds of bodies so it can model things like Neptune and Pluto's orbital resonance, the Jovian Moons, horseshoe orbits (e.g.Cruithne) etc. Not open source but the writer used to occassionally post here.

http://www.shatters.net/celestia/index.html
Celestia has been mentioned above and while I really like it I thiink it is mucho more a visual modeler than an actual simulator. All orbits in Celestia are derived from ephemerides rather than gravitational formulae.

I like Celestia, but I only use it for eye candy, no funky maths. Sounds like ORSA would be the one for that.

There are a few other open source programs of varying complexity:

Orrery

This page has tons of links to resources as well.

SSIM

Has many permutations and a developers version to create scenarios, this might let you play god wtih physics.

Open Universe
Is another package that is open but I don't know anything about it.

There seem to be thousands of programs out there - the programmers must like to take on the physics of orbits as a project!