The story
Popularised by the British mathematician Reginald Allenby in the 1960s, who developed this concept from his daughter, while similar mathematical formulas may have existed earlier in Russia. The number was called "happy" because the iteration to reach 1 was treated like a game of numbers, where reaching 1 is compared to "winning" that game.
Etymology from Wiktionary, CC BY-SA 4.0.