Tolkien doesn't give more than brief notes on the pronunciation of Dwarf names, but given that Dáin is an Old Norse name, the pronunciation 'daa'in' (where 'aa' represents a long 'a' sound, as in 'father') is most likely to be correct.
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Dáin's name comes from Old Norse folklore, in which Dáinn was the name of a deity, who was also a Dwarf or an Elf, depending on the source, and whose name meant 'dead'. Very little is known about this original Dáinn, though stories survive of a cursed sword named Dáinslief ('Dáinn's legacy') that was supposedly made by the Dwarves.