Webster's New World Dictionary and any other dictionary will say that advisor is a perfectly acceptable spelling.
However, if one is bound by the AP Stylebook, as is the case with Mr. Rewrite, go with adviser.
Why does the AP Stylebook call for adviser? Mr. Rewrite doesn't really care. He's just happy that AP is willing to make a decision in the name of consistency.
In Garner's Modern American Usage***, Bryan A. Garner (cue angelic chorus) explains that adviser is the standard spelling. However, he notes that the adjectival form is advisory.