Maybe I can speak to this but not in the context of "canonically supported" information about warp drive.
So whether you take this to mean anything depends on whether you consider the post-Trek-developed concept of Alcubierre-derived "warp" drive concepts to be any authority on how trek warp drives would work. In other words, what I'm about to say is meaningless to this question if you don't think Trek warp drives work anything like our actual real life present day perception of how a warp drive would work.
I just came from that 100 Year Starship Symposium put on by DARPA and NASA-Ames in Orlando last weekend, and while I was there I saw a lecture by Dr. Harold White at NASA Glenn called "Warp Mechanics 101". It was a review of Alcubierre's warp drive metric, including energy density requirements, the mathematical formalism and even some updated mathematics on the subject. That's right, somebody from NASA Glenn is actually working on Alcubierre warp drive mathematics, albeit on his free time and not as part of his NASA-funded research.
But to the point - mass is immaterial to the power you need to create the warp bubble - but there's a caveat. Strictly speaking the only term that appears in the equations is the volume of the space that you need in the center of the bubble to be unperturbed; however, there's a field symmetry paradox wherein it becomes necessary to apply classical Newtonian momentum in the direction of intended travel before you switch on the warp field - because technically the field geometry has to be symmetric, but that would mean without any conventional momentum you wouldn't have a preferential displacement direction, you'd just head one way or the other. You fix that by using a conventional propulsion system to get velocity then go to warp. Incidentally, Trek always did this (probably without knowing why), as you'd generally have to be at impulse to go to warp, you couldn't just go to warp from standstill.
So how does this mean mass is important - well in a roundabout way, the math says that your 'apparent velocity' in a setup like this would be the product of the bubble's superluminal velocity and your Newtonian velocity, and since the acceleration you can impart to a body in space is dependent upon mass - ultimately your final velocity is going to vary inversely with the mass of the ship.
Incidentally, I thought I might pass on a tidbit of information - to say we have no idea how to create warp travel is not entirely accurate. Mathematically speaking it is understood that warping spacetime is done by negative energy densities or exotic matter densities. There have been laboratory scale experiments that have successfully measured negative energy density - see vacuum energy density, casimir forces, etc. These things aren't where they were 15 years ago - mysterious entries in Wikipedia. We know quite a bit more about them. My conversation with Dr. White after his lecture indicated they are trying to setup funding to conduct the first tabletop experiment verifying that spacetime can be warped with vacuum energy.