What is the origin of "melting" in molecular genetics?
The melting temperature (Tm) of a double stranded DNA tract is defined as the temperature at which 50% of the DNA molecules are dissociated into single strands (and 50% form duplexes). Sure, it has a different meaning than in physics... but it is a really common term in molecular biology and I doubt people will stop to use it anytime soon!

There is a vast amount of literature about the calculation of Tm, which is very important also for practical applications, such as designing PCR primers. This page has some explanations and references about the maths behind it. 

I am not sure when the term first appeared, but it has clearly be used since the dawn of molecular biology.

DNA denaturation/renaturation dynamics were first studied in the '60s.

See for instance:

STRAND SEPARATION AND SPECIFIC RECOMBINATION IN DEOXYRIBONUCLEIC ACIDS: BIOLOGICAL STUDIES - J. Marmur and D. Lane - PNAS, 1960

Although Marmur and Lane do not talk about melting temperature, we can find the term already two years later in this 1962 paper:

Effect of Concentration on the Formation of Molecular Hybrids from T4 DNA - Andrzej W. Kozinski and Michael Beer - Biophys J., 1962

Evidently the term was already common there, as they don't care to define it (it's even in the abstract).

Who first used it? I'm not sure, if anyone knows feel free to add a comment or edit the answer.