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?
No, there's no benefit to leaving an empty hand
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:
Cestus| Simple| 1 gp| 1d4| bludgeoning| 1 lb.| Light
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, this can work, 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?
Shields are not gentlemanly. You don't visit the court of a neighboring kingdom in full plate with a two-handed battle axe or a broadsword and shield. You don't attend state dinners with your massive wall of metal. But you can do this with a rapier, or a longsword, or some other medium-length gentleman's weapon. A shield, oddly, is a threat, because it is heavy and bulky and awkward so if you're carrying it people assume you're looking for or at least expecting a fight.
Shields are against the Rules. When you do fight someone, you probably challenged them to a duel, like a proper gentleman. You could do that mounted-on-a-horse-with-a-lance style, in which case yeah, you really need a shield. But you could also do that on foot, in which case the one-handed weapon is going to be the weapon of choice (Though in D&D 2-handers unfortunately are actually better here, unlike in real life).
Shields don't fit into windows, crawl spaces, or your boot. If you're sneaking into a place to do some on-the-down-low stabbing, a shield just isn't gonna do it. Now, as a Paladin you don't sneakily assassinate political figures very often in most campaigns. But you could.
So in a 5e game where looking like a soldier is a bad thing, not using a shield makes sense.
[Warning: I kinda got carried away and cast Wall of Text. Ahem.]
Bottom Line: Yes
In addition to the fact that your character being "worthwhile" depends way more on your group than it does on perfect optimization, I actually can think of a few potential cases where wielding a 1-handed weapon w/o a shield is mechanically beneficial:
To cast spells w/ somatic components, you need a free hand (exception: War Caster feat). If the spell has a non-costly material component, that can be the same hand that holds a spellcasting focus (such as a "shield emblem" holy symbol. However, according to Sage Advice: The Rules of Spellcasting, this only applies when the spell includes a material component. A spell that is only Verbal Somatic requires you to free up a hand.
So if you're wearing a shield and want to cast spells like Cure Wounds, Beacon of Hope, Divine Favor, Dispel Magic, or Elemental Weapon, you need to sheathe your weapon. However, this means you can't take Opportunity Attacks (very well, at least, though I suppose you could Unarmed Strike), and it also ties up your free "interacting with an object" for two turns in a row (Round 1 sheathe, Round 2 draw). By not wearing a shield, you avoid these restrictions.
[If you're a multiclass spellcaster, not having a shield makes it easier to create/access the somatic/material components of spells that can't use your holy symbol as a focus. And if you've picked up somatic reaction spells you avoid having to worry about the logistics of sheathing your sword to cast them.]
Now, a Paladin with a 2-handed weapon might be able to make their somatics freely (take hand off weapon, cast, regrip weapon). But depending on your interpretation of the "interacting with an object" rules, that might be two interactions. This would probably be an "ask your DM" area, but D&D game designer Mike Mearls rules negatively:
can you cast a spell that uses somatic components if you weild a 2 handed weapon or a vertile weapon in 2 hands? nope. -M (Source)
To avoid this, a 2-hander could use a versatile weapon and simply use it 1-handed on those turns where they need to make somatic components. But unless they have the Great Weapon Master feat, the Great Weapon Fighting style probably won't do anything for them that the Dueling style won't do for you.
Of course, War Caster makes this advantage obsolete, but that takes a feat/Ability Score Improvement slot, and if you're a bit MAD like a paladin then those come at a premium.
Lay on Hands
Similar to the above, but not affected by War Caster. You can heal yourself and others of damage and poisons without having to worry about "interacting with object" limitations or giving up your opportunity attacks.
By RAW, you might be able to get away with "touching" a target with your foot or kissing them or something crazy like that. But that's only if your DM is pretty generous. It says Lay on Hands after all (even if ignoring the plural is generally accepted).
Having a free hand can help you move around environments that other Paladins might have trouble with. Need to climb a ladder/rope? That shield will get in the way, and good luck trying to hold on while swinging your greatsword. Dangling from a rain-slicked precipice? You need all the grip you can get. Swimming? Unless you have magic assistance or a natural swim speed, you have disadvantage on all those big fancy great weapons, and a shield will probably slow you down (but a shield-less Dueling style Paladin with a shortsword will do fine).
This is of course assuming that you've somehow addressed the issue of being weighed down by your heavy armor. But that's an issue for all Paladins (unless you go DEX based).
James already mentioned that having a free hand means you can grapple whenever you want. This means that you can prevent a target from moving, or at least get them to waste an action escaping. Perhaps more importantly, a 1-handed weapon Paladin can still attack while grappling. A sword and board has to sheath their sword altogether to grapple, and a 2-handed weapon Paladin can't use their weapon while grappling (unless its versatile, but Great Weapon Fighting style only works when using it 2-handed, while you will still be benefiting from Dueling style).
As a side benefit: having a creature between you and an enemy gives you half cover. This gives you a +2 bonus to AC and to DEX saves. So by grabbing some unfortunate enemy minion and dragging him into position, you can temporarily replicate having a shield, at least vs. ranged attacks from certain angles.
Also consider that the Grappler feat gives you have advantage on attack rolls against a creature you're grappling. You're not only preventing your target from moving, you're getting an attack bonus against them (better chance of hitting, and a better chance of critting and getting some sick smites in). And with Extra Attack at Paladin 5, you can grab and stab in a single turn, and then make two attacks with advantage on the next round if your target hasn't escaped/died. And since advanced smites are bonus action, vocal-only spells, you can pile on one of those as well.
Now, to really take advantage of this, you want to pick a grapple victim that won't die right away (for more advantaged actual attacks per grapple attack substitute), and you want to stop them from escaping. Here's some tips on the escaping point:
(1) hamper their ability checks: if an enemy is frightened or poisoned, they have disadvantage on all ability checks (and also attack rolls). This means that not only will they have a hard time escaping your grapple, they'll have a hard time damaging you as well. Good sources of these conditions are Wrathful Smite (WIS save), the Menacing Attack maneuver (also a WIS save), Ray of Sickness (CON save) from the party wizard, and perhaps some weapon poisons (CON save) [hey, "By Any Means Necessary", right Vengeance Paladins?] You could also have a Bard use some Cutting Words, or have a Warlock Hex whatever you expect the target's better stat is (neither of these have a save, but Cutting Words is negated by deafness and charm immunity).
(2) Boost your own ability checks: getting Expertise from Rogue 1 (or Bard 3) and putting it in Athletics will work wonders, doubling your proficiency bonus on that skill (which is used for initiating grapples and is an option for defending against them). If you manage to find one of the higher level Belts of Giant Strength or a Manual of Gainful Exercise, that will be tremendous. A Bard giving you Bardic Inspiration and/or a Cleric/Druid giving you Guidance will help (but the latter should really only be used if you have time to prepare right before an ambush). Enhance Ability (Bull's Strength) cast on you will give you advantage on your grapple checks. So will Rage from a Barbarian dip, but that prevents you from casting/concentrating on spells. Perhaps the best solution is to get the party Wizard to cast Enlarge on you.
Because not only will that give advantage to your grappling, it will let you grapple adult dragons. Serve with a Strength Hex (to impose check disadvantage w/o challenging the Legendary Resistance and their generally good saving throws) and season to taste with liberal amounts of Divine Smite.