I think the best evidence supporting TIGHAR's Nikumaroro/Gardner Island Hypothesis is the sextant case discovered on the island in close proximity to the unidentified skeleton.
We don't know what happened to the sextant case or the skeleton, so they're only know from the original reports from the British colonial authorities at the time.
This is the text of the radio-telegram mentioning the sextant serial numbers:
Some months ago working party on Gardner discovered
human skull – this was buried and I only recently heard about it.
Thorough search has now produced more bones ( including lower jaw )
part of a shoe a bottle and a sextant box. I would appear that
(a) Skeleton is possibly that of a woman, b) Shoe was a womans and
(probably size 10, c) Sextant box has two numbers on it 3500
(stencilled ) and 1542 – sextant being old fashioned and probably
(painted over with black enamel.
Bones look more than four years old to me but there seems to be very
slight chance that this may be remains of Amelia Earhardt. If United
States authorities find that above evidence fits into general
description, perhaps they could supply some dental information as many
teeth are intact. Am holding latest finds for present but have not
exhumed skull.
There is no local indication that this discovery is related to wreck
of the "Norwich City".
Gallagher.
The notable thing to me is that the sextant case had two different serial numbers on it.
A scan of the original document can be found here:
https://tighar.org/Projects/Earhart/Archives/Documents/Taraw...
The two serial numbers fairly conclusively identify the sextant as a Brandis sextant that had been surplussed by the U.S. Navy sometime after 1918.
This page has more information about the sextant:
https://tighar.org/Projects/Earhart/Archives/Research/Bullet....
If the sextant post-dates 1918, then it and the associated skeleton are either associated with the 1929 wreck of the Norwich City or they belong to the survivor of some other accident. Given the location of the island where the sextant and skeleton were located, it seems unlikely that they would be associated with the Norwich City.
The only other loss in that time period in that part of the Pacific Ocean is, as I understand it, Amelia Earhart and Fred Noonan.