Best mid-range zoom (faster than f3.5) for Sony A57 up to 500$?

Older Sigma 17-70 f2.8-4.0 do have optical stabilization, but they seem quite hard to find used.

In my opinion, the Sigma 17-70 and Tamron 17-50 are the two best options for you. Sigma has a silent motor and a bit more reach, but with a non constant aperture. Tamrom is noisier, with a shorter range, but a constant and larger aperture.
And the tamron is cheaper than the sigma.
Not that easy to choose between the two