What is so special about positive hydrogen ions that they are used in pH calculation?
What we're really talking about with the acidity of a solution is the proportion of water which is dissociated into hydroxide $OH^-$ or hydronium ions $H_3O^+$ (hydronium is usually a more accurate representation than a free proton).

Other ions do affect the acidity of a solution, but by inducing water to dissociate. Check out the example on the wiki page. Pure water alone has a $pH$, because of spontaneous dissociation of some water molecules into ions. The $pH$ of pure water is 7, because the $pK_a$ of water as a weak acid is 14.

Basically, water is the special thing, and its constituent ions happen to be hydroxide and hydrogen / hydronium. Those ions are what we measure with a $pH$ electrode, so the scales are named after them. Water has a concentration around 55 M, vastly predominating in aqueous solutions.