To remove the button from the inside rearview mirror base you insert a flat blade screwdriver in to the mirror base from the bottom to release the retainer. If the mirror is still installed on the windshild then after you release the retainer move the mirror assembly upward to slide it off the botton. If the mirror has come off the window once you have released the retainer you may need a hammer, punch and an extra set of hands to knock the bottom out from the mirror base.
Are you referring to the "button" for the inside rear view mirror which is glued to the windshield of your 1996 Chevrolet Blazer? If so, the best way to remove it is using a heat gun and a putty knife. You will need to apply some heat (not to much) to the windshield and then pop the "button" off with a putty knife.