Yes, I would start by checking battery connections as well as ground connections at the engine.
If everything clean and tight, I would rig a volt meter to the OUT side of the ignition solenoid. See if it is dropping power when this happens.
If so, see if the signal wire from ignition switch to the ignition solenoid is dropping, of if it is a problem with the solenoid.
BTW, other symptoms of this would be for other things like the dash HVAC fan to cease working when power out of the ignition solenoid is failing. Is that happening?