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Given a set of point masses of known mass with known positions and velocity, what is the best way to calculate the angular momentum of the entire system?

The angular momentum of a single point mass is the cross product of the position vector $\vec r$ and the momentum of the point mass, however the position vector $\vec r$ can be chosen arbitrarily for a single point mass. When concerned with a set of point masses, should I select the center of mass as the origin for the position vector?

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There is no unique value for the angular momentum. You can choose any point and calculate the angular momentum about that point.

However for some arbitrary point the angular momentum will generally not be constant because you will be working in a non-inertial frame. The advantage of choosing the centre of mass as the reference point is that this selects a frame that is inertial so the angular momentum will be constant, and this generally makes analysing the system much simpler.

Unless you have a very good reason to do otherwise you should choose the center of mass as the origin for the position vector.

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