हर हर महादेव
Mahadev throat become Blue(NeelKanth) by swallowing the deadly poison that came up during the churning of “Kshir Sagar” or the milky ocean(Samudra Manthan). Hence, He held it in His neck, which turned blue due to the effect of poison.
14th shloka of Shivmahimna Stotram says:
विधेयस्याऽऽसीद् यस्त्रिनयन विषं संहृतवतः।
स कल्माषः कण्ठे तव न कुरुते न श्रियमहो
विकारोऽपि श्लाघ्यो भुवन–भय– भङ्ग– व्यसनिनः॥“
which means “O three eyed Lord, when the poison came up through the churning of the ocean by the gods and demons, they were all aghast with fear as if the untimely end of all creation was imminent. In your kindness, you drank all the poison that still makes your throat blue. O Lord, even this blue mark does but increase your glory. What is apparently a blemish becomes an ornament in one intent on ridding the world of fear”.