One of my players is playing a mounted character, and is claiming that since a lance is a two handed weapon, that it deals double Strength damage, even when wielded one handed while mounted. While this doesn't make sense, does it follow the rules as written?
The lance does double damage if used from the back of a charging mount.
This has nothing to do with the separate rule that if you wield a one-handed weapon two-handed, or a two-handed weapon, it does 1 1/2 x STR damage (not 2x).
I will note that in Hero Lab it correctly simulates this behavior. My cavalier with a +4 STR mod, when mounted, can hold in one hand for d8+4 or two for d8+6. If I am charging, this becomes 2d8+8 one-handed or 2d8+12 two-handed.
Furthermore, this isn't some strange superheroic fighting style - before the development of the vamplate, holding on with both hands was really the only way to employ a spear or lance on horseback. Also note that lances aren't fragile or designed to break (jousting lances perhaps, but not ones made for war).
The Lance has this special quality:
Note that wielding the lance one-handed is optional. You can, not you must.
The Ride skill gives you the following benefit:
So yes, RAW you can wield a lance with two hands, while guiding your mount with your knees.
However to do so, you must still meet all the normal requirements of a two-handed weapon. So, no shield. You must be guiding your mount with your knees to have the weapon ready to strike. Etc.
Here's what the rules say about two-handed weapons:
Note that the higher strength bonus is a function of wielding a weapon two-handed, not of using a two-handed weapon. So wielding a two-handed weapon one-handed (as can be done with the lance) doesn't grant a higher strength bonus (if he wants to use a shield, for example).
(And to clarify: One-handed vs. two-handed changes the strength bonus from 1x to 1.5x. The double damage from using a lance on a charging mount is always present, whether wielded in one or both hands)