I have on the mirror modifier and I only need to select a point one side without deleting the mirror modifier. How does one accomplish such a task?
Your question conflicts with the Mirror Modifier purpose. It is used to mirror your model in a given direction to reduce workflow or give symmetry to your model, you cannot select a vertex of the mirrored part.
Simply apply the Mirror Modifier to your model, and then you can edit one side and it will not affect the other, because you applied the modifier and each side it's independent.