Abstract: We show that puncture data for quasi-circular binary black hole orbits allow a special gauge choice that realizes some of the necessary conditions for the existence of an approximate helical Killing vector field. Introducing free parameters for the lapse at the punctures we can satisfy the condition that the Komar and ADM mass agree at spatial infinity and at each puncture. Several other conditions for an approximate Killing vector are then automatically satisfied. We also construct a sequence of binary black hole puncture data derived under the assumptions (i) that the ADM mass of each puncture as measured in the asymptotically flat space at the puncture stays constant along the sequence, and (ii) that the orbits along the sequence are quasi-circular in the sense that the above mentioned necessary conditions for the existence of a helical Killing vector are satisfied. We find that up to numerical accuracy the apparent horizon mass also remains constant along the sequence and that the prediction for the innermost stable circular orbit is similar to what has been found with the effective potential method. With our gauge choice, the 3-metric evolves on a timescale smaller than the orbital timescale. The time derivative of the extrinsic curvature however remains significant. Nevertheless, quasi-circular puncture data are not as far from possessing a helical Killing vector as one might have expected.