The word Brinji means nothing in Tamil, always a reasonable indicator that the recipe is a borrowed one.
Historians believe rice cultivation came to India from China. From here, it spread to the west. All the western words for Rice come either from the Tamil word (Arisi) or from the Sanskrit word (Vrihi). The Tamil Arisi became the Greek Oryza, Latin Oriza, French Ris, Italian Riso, German Reis, Polish ryż and English Rice. But that's not how our dish gets it name. We need to dig deeper.
The Sanskrit Vrihi became the Pashto vriže, Old Persian brizi and Iranian berenj. In central Asia, especially in Iran (Persia), polished white rice is still called berenj. Persians excelled in cooking rice. It is likely that this polished white rice cooked with mild spices caught the fancy of a Tamil nation and came to be called Berenji or Brinji. Flavoured Rice boiled in milk is still called Sheer Berenji. It is possible flavoured Rice boiled in water might have been called just Berenji.
The armies of cooks accompanying the Sultans Moguls taught us fancier versions of cooking rice like the Pulao & Biryani. Rice was always a luxury, till around 50 years back and was cooked only during festive occasions. One such special dish is the Brinji, which is halfway between plain flavoured rice and and a Pulao / Biryani/ Tahari. The only thing common to all is the use of polished rice. The brinji is much older than Biriyani/ Pulao/ Tahiri, which are relatively new entrants to Tamil cuisine. Being older, the brinji has internalised South Indian ingredients like coconut and pepper. It also uses local rice varieties unlike pulao/ Biryani, which call for Basmati. The texture of a cooked Brinji is not as grainy as a pulao/ Biryani.
Brinji has coconut and pepper, but no curd. Biriyani has curd, but no coconut or pepper. Pulao has no coconut, no pepper, no curd, no meat and usually has a single dominating vegetable/ spice. A Tahiri has no coconut, no pepper and no meat and usually has a bunch of vegetables. I'd think the Brinji is the earliest version of Biryani/ pulav cooked in Tamilnadu. The Tamil name for bay leaves - Brinji elai ( Brinji Leaf) come from the fact they were invariably used in a brinji! Down south, in places like Madurai, the heart of Tamil culture, Brinji is still more famous than the Biryani/ Pulao. True to its roots, the Brinji is paired with kuruma, another central Asian dish. This is not the Korma of the north cooked with dairy, but the south Indian version with coconut taking place of milk.