What is the proper term for lines of constant volume and constant enthalpy on a thermodynamic diagram?
But what do you call a line of constant volume? An Isochore?


Correct.


  And what do you call a line of constant enthalpy?


The process is isenthalpic. More than one engineering/thermodynamic text I found by Googling indicates they are simply referred to as "isenthalpic lines." See this for example.