Re: what gives a stringer its strength
Without getting into a dissertation on structural engineering you should provide a few more details on the type, size, weight and powerplant of the boat. That'll give us a good start to try to help out.
For the most part the core serve two functions in a stringer, to keep the opposing reinforcements apart and correctly oriented to one another. This is important to ensure that the "I-Beam" affect of the structure remains intact under varying loads of compression and tension. A void structure w/no core will provide, as you guessed, absolutely nothing to the structural integrity of the stringer. Foam us used in these structures to provide at least some integrity and help prevent sheering stresses from compromising the part under load. Wood provides much better structural support for the part than foam but as you know is heavier and more prone to rot when exposed to water over long periods of time. Most of the void structures I've seen on racing boats are reinforced with carbon fiber or other high tech fibers. They are also totally enclosed as in a pipe section which provides the added structural integrity.
A void stringer will have to be carefully engineered for attaching points for engines and equipment such as high density inserts and backing plates. Yes they will be made up of heavier laminations of glass & resin which besides cost will also add weight. In essense layering on more glass and resin will create a core but not one that spans opposing sides of the stringer.
If this is a small runabout restoration I'd look seriously at making your stringers out of marine ply or any other good grade ply, encapsulated in epoxy then glassed into place on a bedding of thickened epoxy glue. You'll only need a layer or two (depending on engine etc.) of 18 oz. biaxial fabrics over the stringers and tab them to the hull with overlapping layers of biaxial tape. Make nice gentle radii of your fillets from the stringer to the hull to allow the fabrics to curve gently in the transition.
Here's a good primer on foam core construction from the Diab group (mfgrs of divinycell).
http://www.boatdesign.net/articles/foam-core/index.htm
Pictures would help too. Good luck with your project.