I think it is a negligible thickness. Most depths are measured from landmarks at the skull surface and when you infuse or inject it matches atlas depths.

Super late to the party, but for anyone who finds this question from a search: if by "dorsal side" you mean the side (lateral aspect) of the skull, I don't have a good answer for you. If you mean the dorsal surface of the skull, such as over primary motor cortex (for example), within ~1-3 mm of the midline, then a rough estimate is 0.25-0.30 mm. I recall seeing some measurements in a preprint for Craniobot, a stereotaxic surgery robot. You might want to make a few measurements yourself, at the appropriate age range and skull location.

More to the point: the well-known Paxinos atlas uses "DV" (dorsoventral) coordinates measured from the surface of the skull at bregma. So if you find your target in the Paxinos atlas, and wish to find coordinates for your stereotaxic instrument, the key step is to make sure your surgery setup allows you to bring the mouse skull into the same alignment as used in Paxinos: that involves carefully leveling the bregma and lambda landmarks.

The thickness of the skull does vary somewhat across the dorsal surface, and also varies with age and sex of the mice.

Personally, I prefer to define injection depth from the surface of the brain: then it is not quite as important to achieve "Paxinos alignment" prior to injection. You can just measure AP and ML coordinates relative to bregma on a roughly-level mouse skull, drill a small burrhole, and lower the pipette to the center of the burrhole. Under the stereoscope, visualize the pipette tip and lower it until it just touches the surface of the brain, zero the z-axis control, and lower the pipette the desired distance. Caveat: it can be difficult to see the surface of the brain consistently after drilling a burrhole, but I find that the error is small compared to the size of most injection targets. Certainly for any neocortical injections, this works well.

If you are aiming for very deep brain structures, then the exact angle of the skull in the setup matters much, much more (as compared to e.g. neocortex targets). In that case, aligning the skull to a consistent reference frame is critical. And while it's fine to start with Paxinos coordinates, my advice is to do a series of iterative test injections, using a cheap dye (e.g. fast green). Immediately after injection, kill the mouse and inspect the brain (e.g. by making fresh brain slices and putting them on slides rapidly): assuming the dye missed the intended target, make a small correction and try again. Within a few hours you should have found the coordinates/approach that works well in your hands, and those numbers may well differ substantially from what the atlas would suggest. If you take this approach, you're free to use skull or brain surface as the reference, whichever works best for you.