Is it feasible to store adequate thermal solar heat overnight?
It would almost undoubtedly be a better use of your available money, time, and space to add insulation, not a complicated solar heat storage system, which after all, will do nothing for you half the year.

Start with your basement and insulate the walls and the rim joists. Then move to your attic and air seal the floor and add about 12-16 inches of cellulose or fiberglass. That should help tremendously.

Here are some articles to help you get started on the right track: (greenbuildingadvisor.com and finehomebuilding.com) 


All about basements
How to insulate a basement wall
Insulating rim joists
Air sealing an attic
How to insulate and air seal an attic hatch


After that, keep air sealing everything, and consider replacing your windows if they're on the way out for other reasons too (rotting, impossible to open or close, etc).

Doing all of these things should cut your heating AND cooling bills by 50% or more.