The Sierpinski Problem is sometimes stated as the two numbers found by Selfridge in 1962 are the two smallest Sierpinski numbers. (78,557 and 271,129). For example, this page says "The Sierpinski problem is to show that these indeed are the first two Sierpinski numbers"
http://www.csm.astate.edu/~wpaulsen/.../mazeisol.html
So continuing the project to confirm the second smallest Sierpinski number would be a natural extension.
Sierpinski's work was probably inspired by Riesel's work that proved the same thing as Sierpinski with the "+" changed to "-". The smallest Riesel number is thought to be 509,203. The Proth.exe program can check primality in these numbers, too, so it would be a similar project to check whether this is the smallest Riesel number.
Bier numbers are numbers that are simultaneously Riesel and Sierpinski numbers. The smallest known Brier number I've seen, found in Jan 2000, is
878503122374924101526292469
http://www.glasgowg43.freeserve.co.uk/brier2.htm
Confirming that would probably keep the project going into the era of Petahertz processore.