I've been looking through Poor's (MIT has all but the first few) and here's what I found so far about the Erie Weehawken Branch and surrounding area (more to come once I borrow other volumes):
The New York & Fort Lee RR was projected in 1870 and about 2 miles were operational in 1871. In 1881, it was the Weehawken, New York & Fort Lee, and it was 5.105 miles. Maybe this was Poor's way of showing the Weehawken Branch with the NY&FL. The NY&FL was 0.227 miles long in 1885 and 1886, and disappears in 1894.
Between 1881 and 1885 the Weehawken Branch was built. Apparently it was leased in 1868 (maybe it was chartered as the Docks Connecting RR). It was 3.312 miles long in 1885 and 1886, and 3.45 miles in 1894.
The NJ Junction RR is in 1894, but not 1886. From 1894 part of it was used by the Erie under contract. It was 4.35 miles in 1894 and 4.51 miles in 1897.
In 1897 (but not 1894) the entry under Erie for trackage rights on the NJJ appears as 3.02 miles. At the same time, the Weehawken Branch goes down to 0.43 miles.
The mileages of the Weehawken Branch, NJJ, and Erie trackage rights on NJJ (0.43/4.51/3.02) stayed constant in 1897, 1898, 1901, and 1902.
In 1912 the Weehawken Branch was 0.367 miles. The Erie trackage rights on the NJJ was 3.020, and the NJJ was 4.44. Also in 1912 the New Jersey Shore Line RR (0.85 miles, NYC owned, Erie operated, Weehawken to Shadyside) appears, as does the Erie Terminals RR (1.27 miles, Erie controlled, at Edgewater, successor to Edgewater & Fort Lee RR (Weehawken to Fort Lee)). This stays the same in 1914, except that the Weehawken Branch is instead the Docks Connecting RR. In 1917 the Weehawken Branch is simply an unnamed branch. The New Jersey Shore Line has been absorbed into the NJJ, with the added text being 0.84 miles from West New York to Shady Side.
So, to summarize:
The New York & Fort Lee came first, but it's unclear exactly where it was open. Next was the Weehawken Branch (the original alignment through Hoboken). This completed a line from the tunnel to the Delaware & Hudson Canal docks at Weehawken (later New York, Lake Erie & Western Docks & Improvement Co).
Soon came the New Jersey Junction, NYC operated. The Erie abandoned its alignment through Hoboken and got trackage rights on the NJJ. After the Erie took over the NYS&W, they wanted to connect the Erie Terminals to Weehawken, and the NYC built the line for them. There's a long thread about this in the NYC forum at railfan.net.
Does anyone know the location of the former Hoboken passenger station on the NJJ?
By the way, what part of New Jersey law contains the railroad charters? Is there a way to get this online?