Can ionospheric plasma disturbances effect the weather?
Apparently it is possible for the ionosphere to influence the conditions in the lower atmosphere. 

The following are relatively recent results from a paper in 2009, and a summary and up-to-date state-of-the-art can be found in this paywalled, upcoming chapter in a book.

As it turns out, at least in the arctic regions, the ionosphere can have quite an impact on the temperatures.

Geomagnetic storms provide large numbers of energetic electrons, which follow the magnetic field lines. As the magnetic field lines only intersect with the atmosphere in the polar regions, those electrons will preferentially precipicate there. 
What follows is Ozone destruction and generation of $NO_x$-species in the Mesosphere and Stratosphere. $NO_x$ molecules are classical long-lived Ozone killers.

The Ozone depletion leads to direct heating of the Polar surface of up to $\Delta T \approx 4.5 K$ by increased UV irradiation. Furthermore, the $NO_x$ can be transported towards lower latitudes, and geostorm induced chemistry seems to have influenced mid-latitude Ozone concentrations measurable in the past for short times. A nice summary also in the Science case for the EISCAT3D-radar (not paywalled, and one can skip directly to page 7 where those effects are described and further referenced). 

Those indirect effects are intriguing in that, while some might argue that the ionosphere cannot have any influence on the troposphere, it is forgotten that the radiative energy balance links the whole atmosphere. And it is this balance that can be altered by geostorm induced ozone destruction in the middle atmosphere. It's not exactly what OP was going for (no rise in lightning strikes in the troposphere), but the ionosphere knows their own version of lightning , called Sprites (also a more recent finding in research), which are sure to appear with those events.

Keep in mind that this is still ongoing research.
While a correlation of geomagnetic storms with changing polar temperatures has been already noted in the past, and seems undisputed, the exact mechanism responsible for the correlation is still under scrutiny. Personally I feel there must be more to it, as $4K$ temperature difference just from increased UV irradiation seems a bit much.