standard model - What's with the very slightly larger mass of the neutron compared to the proton?

Neutron mass: 1.008664 u

Proton mass: 1.007276 u

Why the discrepancy?

On a related note, how does one go about measuring the mass of a neutron or proton, anyway?

Answer

Masses and coupling between quarks are free parameters in the standard model, so there is not real explanation to that fact.

About the measurment: you can have a look at this wikipedia article about Penning traps which are devices used for precision measurements for nucleus. Through the cyclotron frequency (Larmor factor) we can obtain the mass of the particle.

Edit: "A neutron is a proton + an electron" is a common answer to this question, but it is a totally invalid reasoning.

Both protons and neutrons are made of three quarks. The mass of the quarks is not known with enough precision, and even more important (and that's a why for the masses of the quarks), the interaction between them is responsible for the mass value to a much larger extend.