Burlington is a bit too Republican leaning to serve as a good bellweather for New Jersey. Bergen used to be too Republican, but it's much closer now to mirroring the statewide results as a whole. (Corzine won it by 13.4, a bit better than his statewide; Menendez won it by 8.3, almost imperceptibly worse than his statewide.) If I had to pick, I'd pick Bergen.
As a bonus, Bergen has a little bit of every kind of New Jersey town (save for the shore resort type): the northern super-rich commuter exurbs, the small towns like Ridgewood (which has a beautiful little "downtown" area), the ethnic Italian suburban-style towns of south Bergen, even the "big city" towns along the Hudson, like Fort Lee. It's a great cross section.
I'll also throw Gloucester County into the mix. Recently, Gloucester has been pretty close to approximating statewide results. Corzine won it by 10.2, Kerry won it by 5.3, Menendez won it by 7.8, and Lautenberg won it by 13.4.
That spread is what allows it to serve as a bellweather, there really aren't many counties like it, especially not in NJ.