Assume someone wants to fly a General Aviation aircraft from North America to Hawaii. Which airport on the U.S. mainland (including Alaska and its peninsula) or Canada is closest to the nearest airport in the Hawaiian islands?
based on google San Fran to Hawaii is around 2300 miles. Looking at a map it would seem LA or San Diego is closer but they are not based on what I found.
Poking at it a bit, it’s going to be something like San Francisco to Hilo, at about 2300 miles. My first guess had been either San Diego or Cabo San Lucas (You said NA) but those are about 2500 and 2900 miles.
Using Google Earth, I get 2,282 miles from Lihau Airport on Kauai to Cold Bay Airport on the Alaska Peninsula, so it’s going to be a near thing. It was difficult to use the tool over such a distance.
Interestingly, Kure Atoll in Hawaii is only 1,576 miles from the nearest spot in Alaskaon Amatignak Island in the Aleutians. But Kure only has a rudimentary airstrip, and Amatignak is not on the mainland.
I don’t think any of the other islands in Hawaii’s northeastern chain have an airstrip. Of course there is an airbase on Midway but although that’s in the Hawaiian chain it is not part of the state of Hawaii. Midway would probably the closest airfield in the Hawaiian chain to the Alaskan Peninsula.
It looks like there is a 2000’ airstrip on the SW side of the Ft. Bragg area that is something like 18 miles closer to Hilo than SFO. But it looks disused and has Xs marked across it. You could probably get away with using it, but, then again, maybe not.
Using an online calculator, I get a distance of 2,303 miles between Lihau and Cold Bay.
Checking, it seems to be available to private use with permission.
Using the same calculator, I get a distance of 2,302 miles from Hilo Airport to the Ft. Bragg Airport. The difference between it and Lihue/Cold Bay would be within the margin of error of the measurements, depending on where on the airfield you measure the distance from.
If you wanted to use actual public airports Lihue/Cold Bay would be closer than Hilo/SFO.
I see a Little River airport (LLR) SE of Mendocino that, when I draw a straight (geodesic) line to Hilo shows as 2297.
The calculator gives me 2,298 for that one.
I see you Little River-Hilo, and raise you Lihue-King Cove Airport (south of Cold Bay), which gives a distance of 2,296 miles. 
PS. This should be Lihue, not Lihau.
I have a book on flying (delivering) new General Aviation aircraft to the Pacific and Australia. (And to Europe). They mention flying to Honolulu from San Francisco, including even the smaller 4-seaters. Going to Hawaii or Greenland, you need a special permit since they (Canada and USA) discourage (ban?) over ocean long distance flights by single-engine aircraft. The book mentioned they fly in convoys in radio contact. They also get a special permit to take off 25% over max gross weight (the nice long runways are handy then), and the majority of that weight besides the pilot is fuel in a special tank in the cabin. The longest leg, as mentioned, was to Hawaii, a little less than 2,400 miles. then island hop to Australia or NZ.
One story I read mentioned how they were flying that leg and a pilot reached over to get a drink from the ice cooler and his hand got stuck. This was funny until it started going numb. They were seriously considering ditching two aircraft since they would never ditch one pilot by himself - but finally he got his hand free.
I wonder what the longest distance (single GCA vector) one can fly entirely over water is. That line from (about) Karachi to (about) Petropavlovsk would be a pretty long trip. It also looks like it might be possible to fly straight from NYC to Perth without crossing land.
Cold Harbor to Lihue looks to me like 2299.4 statute miles according to Google Earth. I measured from the midpoint of the longest runways. Another airport on Kauai is closer: Princeville Airport (2282.8 miles from Cold Harbor), but it’s private and permission is required to use it.
I found four routes from California that are very close to the same distance:
2291.6 statute miles Hana (Maui) to Lofty Redwoods (near Anchor Bay, CA; private airstrip, permission required)
2292.5 statute miles Hilo International to Lofty Redwoods
2293.0 statute miles Hana to Ocean Ridge Airport (near Gualala, CA; open to the public)
2293.8 statute miles Hilo to Ocean Ridge
Princeville to King Cove is slightly shorter at 2280 miles. (And that should be Cold Bay instead of Cold Harbor)
FYI, for standardization I’m using the airport coordinates given by AirNav.com and a calculator likethis one.
I got a better one. As it turns out, French Frigate Shoals in the northwestern Hawaiian archipelago has an airport. French Frigate Shoals to King Cove is just 2,190 miles.
N/M
The OP is somewhat ambiguous in referring to both “Hawaii” and the “Hawaiian Islands.”
Both French Frigate Shoals and Midway are part of the Hawaiian Islands. However, French Frigate Shoals is part of the state of Hawaii, while Midway is not, being an unincorporated federal territory.
If the state of Hawaii is meant, then French Frigate Shoals wins. If Midway is allowed, however, it beats FFS easily. Henderson Fieldon Midway to King Cove is just 1,943 miles.
I don’t know what map you guys are using, but it looks like a short hop from either Alaska or Texas.
A great tool for this type of question is the Great Circle Mapper website