Here's what I know about one-handing IRL that could be applied to D&D:
Generally the point of one-handing is to maintain better balance and such. You have a hand free, it does NOT sit on your hip, it does important shit.
Some important shit it could do:
Formidable bonuses to balance checks. I know balance is useless blah blah, don't care. If it's useless that means you can apply it in obscene amounts and no optimizer will call it broken, thereby freeing you from balance-killers for several mid levels (Where a lot of BFC is coming into play but flight isn't on everyone yet).
Stand from prone as a swift/free action with no AoO. Having an arm free gets you off the ground faster.
Tumble full speed, the ability to tuck one arm under and such is very different from fumbling with a metal shield or a two-handed weapon. Climb while wielding a weapon. Get free attacks any time you make a disarm, grapple, trip, bull rush, overrun, because you have a hand free.
Get goofy shit like eye pokes. Get goofier shit like turning someone's helmet around or pulling their cloak over their head so they can't fucking see. Get super goofy shit like unlacing their armor or snatching their spell component bag (Free disarms against non-weapons every round?). NOBODY uses the the "snatch item" feature of disarm, when in fact it's probably the most interesting option.
In the MUD I play warriors get a skill called "Dirty tactics" that has a random chance to trigger and inflicts random status conditions with 1 minute durations. Shit like half speed (Stomps on your foot), blindness (Gouges you in the eye), silence (Punches you in the mouth), etc. There are tons of status conditions in D&D, inflict them with you free hand.
Get +1 AoO every round.
The suggestions above include IRL benefits of one-handing and some random fantasy shit.