This is a crucial aspect of Stern-Gerlach which is usually omitted from simple descriptions. To see the spin splitting cleanly, you want a massive neutral atom with an unpaired electron. Silver has an unpaired 5s electron, and all others are paired. The 5s electron is in a zero orbital angular momentum state, so as far as the magnetic response is concerned, only the spin changes in different fields.
You use a massive neutral object containing an unpaired S-electron, so that you only split the spin state in the B-field gradient, you don't muck around with orbital magnetic moment inside the atom, or for the particle as a whole. The rest of the silver atom is there to make sure that the only difference in deflection is due to the spin of the outer electron only.
For free electrons, the spin and orbital magnetic moment are related by the Dirac equation, and you can't make the splitting caused by the two different. This is the weird degeneracy in Landau levels, where the spin-splitting is exactly equal to the orbital splitting. This means that the momentum deflections of an electron beam by a magnetic field gradient is always going to be comparable to the deflection due to the different spin states. So you don't split an electron beam cleanly into spin states using magnetic field gradients.
The inner shells of the silver are paired up, so that the electrons are rigid--- you can't mix the electron states in any of the inner orbitals. But the outer electron spin-1/2 makes two degenerate states (ignoring some infinitesimal coupling to the nuclear spin), so the Silver atom ends up being just a massive neutral object with a magnetic moment equal to the magnetic moment due to the spin of a free electron.