Checking around, I find that aside from fringe sites, the only places the topic seems to be discussed is in technical papers that go well over my head.
For example, there's a pdf at
http://www.ann-geophys.net/23/3487/2005/angeo-23-3487-2005.pdf titled "Solar eclipse effect on geomagnetic induction parameters," which says things like "Bencze et al. (2005) studied Pc3 pulsations during the 1999 eclipse. They found a localised decrease in their activity, concentrated at the dark spot. The amplitude decrease amounted to about 50 percent. Leaving the totality zone, the effect disappeared rather quickly. Moreover, the decrease moved in the same direction as the dark spot, and the velocities were similar, too. This effect was interpreted as a switch-off of the FLR mechanism due to a change in the eigenperiod of the geomagnetic field lines in consequence of a loss of charged particles along the field line. Since without solar electromagnetic radiation, the electron density decreases in the lower part of the ionosphere and consequently upward transport of charged particles weakens and the density of charged particles decreases both in the upper ionosphere and plasmasphere."
I also found an abstract at
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APCPCS000899000001000345000001&idtype=cvips&gifs=yes of a paper titled "HF Propagation Measurements During the 29 March 2006 Total Solar Eclipse in Turkey."
As nearly as I can make out, the idea seems to be that the sudden cutting off of solar radiation during an eclipse affects the earth's ionosphere and magnetosphere, and that those changes in turn have an impact on the earth's magnetic field which is "concentrated at the dark spot."
Linking that to gravitational anomalies is something else again, though -- which is why it's important to find out first whether those gravitational anomalies are real or not, and therefore whether it's worth scientists spending time trying to figure them out.
I should say, by the way, that I typically read fringe scientific theories the same way I read most conspiracy theories -- as stuff that's interesting even if not yet ready for prime time. The failure of 20th century science to actually explain gravitation in any meaningful way is one of those major, potentially paradigm-shifting problems lurking out there at the margins, which is why there's interest in anything that promises to offer fresh clues.