With Mathematica 10 you can give each point a color of its own with
pts = Table[{RandomColor[],
Point[{RandomReal[], RandomReal[], RandomReal[]}]}, {20}];
In earlier version you may have to type a little more:
pts = Table[{RGBColor[RandomReal[], RandomReal[], RandomReal[]],
Point[{RandomReal[], RandomReal[], RandomReal[]}]}, {20}];
