It's the artifacts that get the most press, but in my layperson's opinion it's not the artifacts recovered that constitute the best evidence for the theory but rather archival evidence that's been discovered over the last 20 or so years.
Here are a couple of links that I personally found most compelling: http://tighar.org/Projects/Earhart/Archives/Research/Bulleti.... I posted this link to both Hacker News and r/history a few months ago under the title "What happened to Amelia Earhart: The sextant box with two serial numbers".
The second link is: http://tighar.org/Projects/Earhart/Archives/Documents/Bones_.... This page contains radio telegram transcripts from the British colonial authorities responsible for administering Nikumaroro in the early 1940s. The part about the sextant and two serial numbers is about half way down the first page.
So anyway, this is just some circumstantial evidence. But it's not the only evidence that TIGHAR has. If you start combing through their website, you'll find they have a bunch of independent circumstantial evidence, none of which is conclusive by itself, but taken together starts to look suspicious.
By TIGHAR's own admission from your link: "We’re trying but, so far, we’ve been unable to find documentation to prove that Brandis 3500 was assigned N.O. number 1542 or that it was owned by Fred Noonan."
The medical examiner claims this is a male skeleton, not less than 45 years of age, probably stocky and a mixed European/pacific heritage.
http://skeptoid.com/blog/2012/03/20/more-amelia-earhart-nons...
If one were cynical, one could conclude that the annual "finds" are just exercises to keep TIGHAR's name in the news.
That said, if they dive down and find the plane they are going to be pretty pleased with themselves. As far as I can tell there aren't any facts that would rule out their theory that Earhart ended up here.
Note: their copyright disclaimer in the footer technically disallows linking to or visiting their site, so click at your own risk! ;)
http://en.wikipedia.org/wiki/Phoenix_Islands_Settlement_Sche...
Earhart disappeared in 1937, since 1938 there were people on Nikumaroro and it seems they didn't report anything that would certainly be easily visible one year later on a 5 x 1.6 miles big island if Earhart and Noonan managed to land and live there.
See also:
So here's the thing -- they did find evidence and they did report it. See my other comment (the one starting with "A lot of people are skeptical of TIGHAR's Nikumaroro hypothesis...")
This starts to look more and more embarrassing in retrospect.
I'm aware of some of the issues regarding continuous RF transmission of basic telemetry while inflight. The bandwidth for all relevant flight info is higher than you'd think, and the receivers over open ocean are sparse. Et cetera.
But still: in hindsight, relative to the high standards we expect from flight systems, the inability to localize a 777 is looking more and more, for lack of a nicer term: stupid.
I'm reminded of the crash of an early Mars lander (Mars Polar Lander, 1999), which didn't have telemetry during EDL, so the cause of the crash could not be fully determined. They pretty much determined after that, that every Mars landing would have telemetry, so we'd at least know what went wrong.
And in the case of Earhart, it was a boundary-pushing world record attempt. ;)