What was special about the first nuclear test, rather than the thousands of others, at least hundreds of which were also in the Nevada desert?
Obviously it's historically significant, and the new forms of matter were first discovered there , so that's why trinitite is named after the site. But 80 years later, wouldn't we expect the other bomb sites to have just as many interesting chemical reactions?
We do. https://en.wikipedia.org/wiki/Trinitite#Similar_materials says "Occasionally, the name trinitite is broadly applied to all glassy residues of nuclear bomb testing, not just the Trinity test" and lists hiroshimaite and kharitonchiki as similar glassy residues from Hiroshima and the Semipalatinsk Test Site, respectively.
The general term for for the fused glass-like material formed during a nuclear test" seems to be "atomsite" - https://www.osti.gov/pages/servlets/purl/3001767 . Some images of atomsite from the Trinity and Semipalatinsk test sites are at http://www.radioaktivitaet.uni-bremen.de/downloads/Pittauero...
As a wild guess, what's special is it might be the easiest to get, with many samples to study.
The Trinity test took place in New Mexico, not Nevada.