Ummm... No. The gravitational influences of the sun and moon (tidal forces) can and do affect the actual value of g (the local gravitational field). With a scale sensitive enough, you can actually measure the changes of g during the course of a day.
The concern isn't about rapid fluctuations, it's about how the local environment can affect readings. The magnetic field of the earth, atmospheric pressure and buoyancy, tidal forces from the sun and moon are not rapid fluctuations, but can all have some tiny effect on the weight of the reference mass. The real question is which of those influences are significant enough to affect the actual reading?
According to Wikipedia the daily gravitational fluctuation caused by the Sun and the Moon is only 0.000002 g0. I'm going to go out on a limb and say that the scale used in the experiment is not capable of detecting this magnitude of gravitational shift.
Also, it is ABSOLUTELY about rapid fluctuations because the scale is tared. When you tare the empty scale, it compensates for all of the magnetic fields and atmospheric pressure and tidal forces, etc... So, these forces would have to change, between the time that the scale is tared and the time that the gnome's weight is calculated to have an effect of the gnome's reading. Here is an example to illustrate: The scale is tared with a 10g weight on it, so the scale resets itself to 0. You place the gnome on the scale next to the 10g weight and take a reading. Does the scale show the weight of the gnome or the weight of the gnome plus 10g? It shows only the weight of the gnome. The weight would have to be removed or more weight added BEFORE the reading was taken for it to be affected. Tidal forces and atmospheric pressure, et al. just don't change enough in 10 seconds to affect the readings, so it's a moot point. Which is actually the WHOLE reason for taring a scale, to make all of those things moot.
Which brings me to Tom's point about all of these wind-swept areas where all of the experiments have been performed. It's Occam's Razor, once again. What's more plausible? That the Earth's gravitation field is causing the weight fluctuations to match the Gnome's position on the Earth or that in EVERY place the the gnome was weighed, there just happened to be a gust of wind that was exactly and correctly altering the readings of the scale to match what is predicted by the gravitational calculations? That's pretty amazing timing by all of that wind in all of those different places on all of those different days, even inside buildings, wouldn't you say?