A 2014 Slate article on the history of the Kiss & Cry notes the term was coined by Finnish official Jane Erkko more than 30 years ago, according to the book Cracked Ice: Figure Skating's Inner World by former skating judge Sonia Bianchetti Garbato. As you might expect, Erkko noticed skaters were kissing and crying in the box, and it became "a joke among the skaters and with Jane," wrote Garbato.
It became more entrenched in 1983 when CBS was broadcasting the 1983 World Figure Skating Championships. Erkko told a producer the area was called "the kiss and cry corner." According to Garbato, the producer wrote down "KISS & CRY" on the map for the TV crew, and that was the first time reactions were broadcast from the Kiss & Cry.