Question

what changes could we make in harmonic oscillator that would double the maximum speed of the oscillating object?

Answer 1

We can, for example, double the amplitude of oscillating objects.
It will increase maximum of potential energy in 4 time, potential energy is proportional to square of amplitude, E=kx^2/2
Therefore, maximum of kinetic energy also will be increased by 4 times, and the maximum speed then will be doubled as E=mv^2/2.