I'm a Malayali, so you should take this with a grain of salt, but I think "Nagaratharin" or "Nagaratharathu" would be correct.
I looked at the Tamil script for the name and it's written with a short a, not the long "a" of Sanskrit "nAga" (snake), so it actually looks like a combination of "nagaram" (from Sanskrit "nagara", meaning "city") and the plural suffix "Ar" ("people") with the "m" of "nagaram" changing to tt before the suffix as per the regular process. So the meaning seems to be something like "people from the city" (Poompuhar?).
May I complement Kannan91's perfectly accurate information by another meaning for nagaram? Here is the entry from Tamil Lexicon:
, n. < நகரம்¹ +. Nāṭ. Cheṭṭi. Loc. 1. The nine Šiva shrines of Chettinad, each of which is worshipped by particular sections of Chetty families; நாட்டுக் கோட்டைச் செட்டிவகுப்பினர் பரம்பரையாய் வணங் கும் ஒன்பது சிவன்கோயில்கள். 2. Exogamous section of Nāṭṭukkōṭṭai Cheṭṭies; நாட்டுக் கோட்டைச்செட்டிகளுள் ஒரு வகுப்பு.
So nagarathaar has come to denote members of one section among nattukkottai chetties.