The article you link to doesn't have the word singlet in it, but if I'm not mistaken it's a singlet under SU(3) flavor symmetry (the one that would be exact if the up, down, and strange quarks all had equal mass).
Generally, if you have two spin 1/2 particles, they transform as some linear combination of a (total) spin 1 and a (total) spin 0 part. If you look at the state for the eta meson, though... hit it with an S^2. The u, ubar, d, dbar, s and sbar parts are all eigenvalues of S^2 with eigenvalue (1/2)(1/2+1) = 3/4. You can check that S^2|eta> = 0.
Recall from QM that if you have two spin 1/2 particles, there are four states (up up, up down, down up, down down). We can instead use the basis of total spin and the total state's S_z. There are 3 states with total spin 1 and 1 state with total spin 0. These are known as the triplet and singlet states, respectively. The eta, being a spin 0 linear combination of "two" spin 1/2 particles (two quarks), could have either spin 1 or spin 0; this particular state, as argued above, is the spin 0 part. That's why it gets the name "singlet".