We demonstrate the effect of single-electron tunneling (SET) through a carbon nanotube quantum dot on its nanomechanical motion. We find that the frequency response and the dissipation of the nanoelectromechanical system to SET strongly depends on the electronic environment of the quantum dot, in particular, on the total dot capacitance and the tunnel coupling to the metal contacts. Our findings suggest that one could achieve quality factors of 10(6) or higher by choosing appropriate gate dielectrics and/or by improving the tunnel coupling to the leads.