When punching an indent into a piece of metal, the residual stress that is concomitant with the indentation implies yielding and so you have a non-linear interaction.
If you still want to do the linear calculation just to see what it would be, you set the potential energy of the Earth-mass system equal to the energy absorbed by your material during impact:
Note here that we are considering the deflection, x, to be non-negligible in comparison to h, but in reality it almost certainly would be in the small-strain (linear) case. F is the maximum force felt during impact, which happens when deformation is a maximum and the mass isn't moving momentarily.
The equation for the maximum force during impact is derived from the small-strain case where the static deflection quantifies the stiffness of the material:
after substituting for F and solving for x, it can be shown that:
All system properties are essentially captured in the static deflection, mg/k. K is the stiffness of the material, which can be thought of as a spring constant. It depends on the shape of the part and for rods loaded axially it is AE/L, where A is the cross sectional area, E is the Young's Modulus, and L is the length of the bar. Estimating K is the hardest part of your problem.
For more information, Ch. 7 of Juvinall's Fundamentals of Machine Component Design is a good starting point.