The original term in Tamil-Malayalam for the island of Sri Lanka is īr̤am /iːɻɐm/ ஈழம் ഈഴം (pronunciation subject to dialectal changes, e.g., [iːjɐm], [iːɭɐm], [iːɭɐ̃] &c.).

It's usually etymologized from Sanskrit siṁhala-, perhaps via a Prakrit reflex sīhaḷa-.

Thus, an early pre-Tamil loan *cī(ya)r̤am ~ *cī(ya)ḷam from this Prakrit sīhaḷa- may be reconstructed. Then, old South Dravidian languages, including Tamil-Malayalam, underwent an irregular, but not uncommon, loss of initial *c-. This loss is attested in both inherited ...

... Dravidian vocabulary as well as Indo-Aryan (early Prakrit) loans. E.g.,
- Proto-Dravidian *cil(l)- (existential negation morpheme) → Tamil-Malayalam ill-, Parji cil-, Konda sil-
- Proto-Dravidian *cir- ('dark, black') → Tamil-Malayalam ir-, Parji cir-, Konda sir-

- Sanskrit sahásra- ('thousand'), sāhasrá- ('thousand-fold') ⟿ Proto-South Dravidian *cāyi(ci)ram → Tamil-Malayalam āyiram, but Kota cāvrm, Kannada sāvira & Toda sōfer retain the initial *c- [t͡ɕ- ~ s-]

Thus, if sīhaḷa- were loaned early enough into pre-Tamil as ...

... *cī(ya)r̤am, it would become īr̤am more-or-less regularly. The source of the loan might have been pre-Sinhalese (i.e., pre-Helu Prakrit). Sanskrit siṁhala- gives hēḷa- there following the well-known Sinhalese sound change of /s > h (> ∅)/, and this hēḷa- no doubt ...

... descends from a pre-Helu Prakrit *sīhaḷa-, perhaps while the ancestors of the Sinhalese were still on mainland India on their way to the island(s) through (South) Dravidian territory.

Okay, the etymology in Sinhalese is probably wrong. Cf. this thread by @adam_farris1: https://twitter.com/adam_farris1/status/1391539671373475841?s=20