How does mass affect simple harmonic motion?
The answer to the question is not really straight forward. I would say the effect of mass on simple harmonic motion that it will execute under given conditions depends on the nature of the "Restoring Force" . And the final characteristic frequency depends on the ration k/m or restoring force constant/ mass. So if the restoring force itself depends on mass then the m's could cancel leading to no dependence on mass at all on the other hand if the restoring force is a constant then the characteristic frequency will depend on mass.
Let me illustrate using examples:-
1)A mass attached to an Ideal spring:
In this case the Characteristic or the Natural Frequency of the system Does depend on the mass. As you can see the restoring force constant i.e. the spring constant does not depend on mass and hence the resulting motion Does depend on mass.
2)A simple Pendulum:
In this case the restoring force constant i.e mg itself depends on mass and in this case it turns out that the ratio of restoring force constant to mass is a constant. Hence the natural frequency Does not depend on mass.
These are just two very simple examples to illustrate my point we could very well have restoring force constants that lets say depend on m^2 and hence the ratio k/m will still depend on mass. So, in general the simple harmonic motion will depend on mass, but in certain special cases where the restoring force constant is directly proportional to mass, the natural frequency turns out to be independent of mass.