Does the speed of an ion affect the current that can flow through a solution?
There is a strong dependence on temperature in conductivity of ionic solutions; see http://www.eutechinst.com/techtips/tech-tips44.htm, for example. However, thermal energy, i.e. molecular motion, is random, and does not favor current flow in the forward (or backward) direction. There are a number of factors that temperature influences:


Viscosity usually changes  inversely with temperature, so ions move more freely at higher temperature.
Ionic solutions have other species than OH- and H+; for example OH3+ [water + hydroxyl radical] is the hydronium ion; see http://en.wikipedia.org/wiki/Hydronium. The amount of each species varies with temperature, as shown in that reference. See also http://en.wikipedia.org/wiki/Self-ionization_of_water. Depending on mass, hydrogen bonding etc. different species have different conductivity.