Hindu Mythology
According to Hindu mythology Lord Shiva has gained this epithet when he consumed the Kalakuta(Poison) that emerged from Samudra Manthan.
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Etymology
Iravatham Mahadevan an Indian epigraphist has recently published an interesting paper How did the ‘great god’ get a ‘blue neck’? a bilingual clue to the Indus Script that gives the origin of the title Neelakanta for Lord Shiva based on Bilingual Parallels.
According to Mahadevan, the phrase Nil Kanta is the phonetic value of the Indus sign pair meaning "great personage" in proto dravidian language. This title when borrowed in to Indo-Aryan as a load word with only minimal phonetic changes has changed its meaning to Nila Kanta which means "Blue Neck" in Indo-Aryan. Concurrently, the meaning of the derived title nīlkanta passed into Indo Aryan as a loan translation, Mahādēva, ‘great god’. Both the loanword and the loan translation refer to Shiva who had emerged as the ‘great god’ in this period.