I would really like my pally to use just ONE weapon, and wield nothing in their other hand (or at least appear to do so), but I don't really see any thing that makes it beneficial at all. Using 2 weapons or a sword and shield just seem all around better. Is there anything that adds benefits to one weapon at all? Or is there an alternative to a shield that isn't an obvious weapon, like a glove or something?
Based on the latest playtest material of 2013.10.14 there is no incentive to leave a hand empty.
And logically, why would there be one? You have two hands, use them. If you miss a hand, just strap a shield to your forearm.
To offer an alternative answer to the accepted one:
Remember that role playing is not all about combat. You can have a character that only uses one hand in battle, sure! And you can have a really good time, and you can look bloody awesome at the same time.
Furthermore, having a free hand could open some possibilities in the narrative during battle, for example manipulating objects, using improvised weapons, or dragging a companion out of danger, whilst still wielding your primary weapon, GM permitting.
If you're looking for a weapon for your off-hand, on the off-chance you need it, you could opt for the Cestus, which is a type of battle glove. Here are the stats for this weapon in D&D Next:
Perhaps your GM will permit you to have a magical cestus, which gives you a bonus to AC, or a bonus to damage with it.
If your character has lost an arm through some horrific accident, or was born one-handed, that's not the end of the adventure for that character. You can still participate in battles; you can still take part in investigations; you can still explore the ruins of that lost temple over there.
Not everything about RP is about min/maxing your character's statistical abilities. It's about telling a story, as a group, and about having a good time.
Speak to your GM and perhaps she will weave something into the campaign for you.
This answer is not specific to the paladin, but could apply equally to a paladin or most other class types.
Yes but only in a specific kind of campaign.
People often fight/fought with a sword and a free hand in real life. Why do they do it?
So in a 5e game where looking like a soldier is a bad thing, not using a shield makes sense.
You should take "Great Weapon Fighting" and then us a versatile weapon like long sword...then you can use it to do extra damage with two hands or just hold it in one if you like how it "looks". Also you will be able to grapple whenever you want.