The basic designation was W for Airborne Early Warning (AEW) and F for Grumman. Up until then, W was a suffix, not a prime mission designation, as in TBM-3W. It indicated that an existing airplane type, the third modification (3) of the first Torpedo Bomber (TB) to be built by Eastern Aircraft (M), had been modified for the AEW mission (W) by substituting a large radar and its cover for the bomb bay. (The 3 actually reflected that it was the third modification of the TBF, since the airplane was originally designed, developed, and produced by Grumman as its first torpedo bomber, with Eastern Aircraft, a division of General Motors, taking over its production from Grumman.)
As an aside, the similarly configured Grumman AF-2W was used for AntiSubmarine Warfare (ASW), not AEW, with the radar being used to detect a surfaced submarine or more likely, its snorkel. It was paired as a hunter with the AF-2S, the killer half of an ASW team, which was equipped and armed to localize and sink the submarine.
Grumman was in the process of replacing the AF-2W and AF-2S with the S2F, a twin-engine airplane that combined the mission equipment and armament of those two airplanes into one. The S for ASW was now a prefix as W would be for AEW.
The S2F's small radar that was housed in a retractable "dustbin" dome under the aft fuselage was optimized for surface surveillance, not AEW. Grumman therefore proposed a minimally modified S2F airframe for the AEW role, with the APS-20 radar and dome mounted above the cockpit rather than under the fuselage, which would have required a much longer landing gear and therefore required a folding vertical fin/rudder to meet the hangar-height limitation. The forward location of the radome meant that the existing S2F wing-fold system that overlapped the fuselage could be retained. The only change required to the airframe therefore was a slightly deeper fuselage with an aft cabin door and the addition of finlets to the horizontal stabilizer to compensate for the sail area forward that was added by the radar dome.
The Navy ordered two prototypes as the WF-1 and assigned them BuNos 133043 and 133044. Grumman accomplished wind tunnel tests and a fuselage mockup before the WF-1 effort was terminated, probably due to budget priorities and the availability of the AD-5W.
Eventually, however, the Navy felt the need to use a bigger, better radar for the AEW mission. The existence of and commonality with the TF-1 Carrier on Board Delivery (COD) variant of the S2F, which had been proposed at the same time as the WF-1, was no doubt one of the deciding factors in the decision to award the program to Grumman.
Incidentally, the COD version of the S2F was designated as T for Trainer rather that something like S2F-1R like the TBM-3R because the majority of the missions it was originally intended to be used for were training. See https://thanlont.blogspot.com/2011/05/carrier-onboard-delivery.html
The resulting WF-2 bore only a family resemblance to the S2F and TF, however. The radar was now mounted above the fuselage, which required the substitution of an H-tail and a reversion to the sto-wing fold system that was a Grumman invention. The capacious TF-1 fuselage was lengthened ahead of the wing, probably to maintain the required center of gravity. Because of uncertainty about the aerodynamic implications of the huge radome, TF BuNo 136792 was modified, except for the fuselage extension and sto-wing, for initial flight test with the radome. (Note the feathered engine.)
In the following picture, the aerodynamic prototype is flying in formation with a production WF-2. Note the different location of the propeller strip versus the pilot's side window, which indicates where the forward fuselage was extended forward.
I can't confirm that the prototype was designated the XTF-1W as Wikipedia currently states. It doesn't seem likely and Larry Webster confirmed that its history card only states TF-1 and C-1A. A very early (but not dated) Grumman S2F brochure describes the AEW and the COD versions as the WF-1and TF-1 respectively. I've also read that the aerodynamic prototype was designated WF-1, with the production airplane therefore being the WF-2. That's definitely bogus.
After Grumman flight test of the aerodynamic prototype, the radome was removed, the TF/C-1 interior installed, and the aircraft utilized as a transport by NAS Quonset Point, RI.
According to Angelo Romano, there were three other topless Tracers that were used for pilot training since field carrier landings and real carrier landings, not to mention catapult launches, were hard on the electronics, not to mention that the radar and guys in back were unnecessary for that requirement.
When Navy aircraft designations were changed in 1962 to be consistent with the Air Force system, the WF-2 became the E-1B. Although it could have been argued that there was no need to reflect a universe in which the WF-1 had not been cancelled, the group in charge of redesignation chose to do so.
For a multi-part history of U.S. Navy AEW by a naval aviator, Steeljaw Scribe, who has forgotten more about the subject than I will ever know, see: