The mass of something and the weight are two separate things, and this must be clear when asking the weight of protons. First of all the mass of an object, or particle in this instance, is the amount of matter in a material. The weight is a measurement of the force of gravity on a material. The mass of a proton is measured as 1.0072766 a.m.u. or 1.6726 x 10-27 kg. A.m.u. stands for the atomic mass unit, which is a very tiny fraction of a gram.
When discussing the weight of a proton, usually it refers to the mass of a proton The weight and mass of a proton, although different for materials are in certain circumstances such as the weightlessness of being in space. The weight is affected, but not the measurement of mass.