I'm a little late to this conversation, but I'm familiar with the problem in question, and I don't think any significant fix has been attempted that I recall seeing.
I don't feel that the hollowness in the underside of the sheet metal lower portion is the main problem, or that filling the void will make the lathe significantly stiffer. The problem I see is that the headstock (a sheet metal box, in this case) isn't rigidly held perpendicular to the bed (the two parallel square tubular steel pieces). Welding a simple base for the base out of some reasonably stiff square tube stock should do the trick. If it were ~2" square stock, two pieces run beneath the bottom like railroad track would likely be a very robust fix AS LONG AS the headstock, especially the end where the spindle protrudes and grabs the workpiece, is very securely mounted to the new rail thing.
This same idea could probably be executed using wood as the main structural part, but would be heavier and less portable. The bench the lathe is on right now (the old metal lathe bench) isn't ideal for a variety of reasons. I would propose building a new bench, or rather a set of legs for it. Something not too far from being a wide-stance sawhorse with just enough platform on top to support the lathe bed would be compact for storage yet sturdy enough for normal use, and would have significantly less horizontal surface area to accumulate wood chips and debris.
Something roughly this shape:http://www.redmondmachinery.com/prodimg/WMH-708352.jpg