Well, let's begin by clarifying the time frame. Gilligan's Island ran from 1964-1967.
Given that, we need to define the scenario under which the Island might be "found." I see a few possibilities.
The easiest way that this might have occurred in real life would be for the castaways to radio a distress message (which is actually a plot point in more than a few episodes). Building a spark gap transmitter is something that could conceivably be done using scavenged parts from the radio we see in the show or by repairing/cannibalizing any marine band radios that might have been part of the Minnow's equipment (scratch built "foxhole radio" receivers were built in WWII POW camps from scavenged parts). Scavenging the alternator from the Minnow's engines to provide power when the batteries ran out and ripping out the boat's wiring for a long antenna would be recommended. If bounced off the ionosphere, at night Ham radio signals can easily skip over the horizon. The real problem to me (recalling my RF electronics classes from 25+ years ago) would be sending out a signal on the proper frequencies (especially without external equipment to calibrate the circuit) where other Hams would be listening and dumb luck that the wave would skip far enough for it to be intercepted at the exact time someone would be listening.
EDIT1: Also note the difficulties determining the location of the Island as /u/ProfShea discusses. Once alerted, Coast Guard ships and planes could use directional antenna to triangulate on the Island, but ideally a rough latitude and longitude would really help in the effort.
EDIT2: As I mentioned here if there was Loran equipment on the Minnow, even damaged, one might build a transmitter to jam Loran signals. This might well attract the attention of any shipping in the area using Loran and would likely be reported back to maritime authorities. It would be very illegal and might actually endanger shipping, but it could conceivably get them rescued.
Gilligan and company can also hope for two types of satellite surveillance. Various Corona spy satellites were in use at the time that had enough resolution to conceivably have made out the structures the castaways built. However, they depended on dropping a film canister from the satellite which would then be retrieved by a plane in flight as it reentered the atmosphere and deployed a parachute. Unless the Island happened to be right next to a target of military importance (i.e. Cuba, a passing naval task force, etc) it is highly unlikely that a spy satellite would be committed to photographing the area.
Alternately, there were weather and earth observation satellites which might have had the resolution to observe the island as a whole, but would not have been able to make out distress fires or signals by the castaways. IIRC, there was actually an episode where a manned spacecraft (a Gemini like capsule I believe) does observe the letters "SOS" spelled out in burning logs but when Gilligan extinguishes one of the logs to spell out "SOL," the astronaut takes it as a friendly greeting as his nickname is "Sol." Sadly, even with the telescope Sol is shown using in the episode, I doubt such resolution would be possible.
That leaves us with terrestrial observation via smoke and fire signals, which would rely on the Island being near a major shipping lane. Without its location, I can't speculate further on that option.
Finally, while the OP posted this light hearted query, during the show's run, the US Coast Guard received about two dozen telegrams from viewers begging a ship be deployed to rescue the castaways. Whether these pleas were genuine or pre-Internet trolls, I will not speculate.
I think half of the problem is determining where giligan and his buddies are... Using a crudely crafted sextant or octant, Gilligan and his buddies could have determined their lattitude by utilizing a local apparent noon measurement.
Determining a longitude is geometrically more tasking, but they could have used simple dead reckoning skills to get an approximate longitude of the island. Simply stated, the boat leaves at X hrs and travels at an approximate speed for 1.5 hours. You'd have a pretty good estimate of where the island is. The captain probably had a good idea of the Lat/Long of their starting point because charts are very exact(this flag pole/building/lighthouse/buoy/etc has a Lat/Long of XX XX XX YYY YY YY)
In addition, systems off the coasts of America had utilized navigational aids such as Loran and Loran C. These systems were not always accurate, but they were precise(measurements were very repeatable). Perhaps someone could talk about how a scientist could make heads or tails of the signals. Other nations had similar products during and following WWII.
The second half of the problem is reporting that location to someone that can do something about it. I don't know how to make a radio, but I imagine crafting a telegraph wouldn't be outside a scientists grasp. The captain, being a licensed merchant mariner at that time, would have known morse code(it's still part of the testing today). "Help.. XX degrees XX minutes XX seconds North YYY degrees YY minutes YY seconds."
Nautical radio signal technology at the time of WWII knew of the effect atmosphere had during night hours; signals can bounce halfway around the world at night whereas during the day they were more limited.
source I studied navigational science and have seen it in action.
As a side note, technology may not have been the issue:
Recurring elements center on one of five primary themes.
The second theme involves visitors (not others who were stranded/shipwrecked) to the "uncharted" island. One challenge to a viewer's suspension of disbelief is the frequency with which the castaways are visited by people who do nothing to assist them. Some have hidden motives for not assisting the castaways.
So the island's location was known. The question might then be Using 1960's technology, how hard would it have been to call for help from Gilligan's Island? Imagine the crew was not shipwrecked, but held hostage/abandoned, which they were if someone knew they were there.
~info wikipedia