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Let's say that my party is fighting a bunch of kobolds outdoors. The kobolds are slinging a bunch of rocks at the party, and one player says, "I ready my action until a rock comes my way, and then I bat it back like a baseball."
Is this allowed? What would it look like mechanically?
The action you described is not really a readied action. It's more like a "colorful" Dodge Action.
Either dodge or a readied action will cost one action - they are equivalent in game mechanics. If the player wants this to be a readied action, there's no reason to correct them on the term.
Disguising the Dodge action as a readied action
You can indulge the player having a little fun like this just by letting him make the roll that you would usually make, because of the disadvantage imposed by Dodge.
Swap your "disadvantage" roll for the complementary roll the player would make. That is, if the kobold's first roll hits, then the player rolls to try to whack the stone away. Success is a roll equal to or greater than:
20 - (target's AC) + (attacker's sling attack bonus)
So far, you haven't really changed the kobold chance to hit at all. The player may not notice the rules are really the same as dodging. After all, they are getting to roll the d20 now.
If you wish to add a little color, then let him hit a kobold on the roll of a 20, but hit a party member with a "foul ball" on a 1. No real advantage over rules-as-written, just a little fun.
Power gaming vs. just finding the fun
A little aside about player intention here. The tone of the question sounds like the player is looking to try something creative during a minor encounter with kobolds, and that sounds fun. If your campaign is a "fun fantasy" style game, and not a gritty, realistic one, then you want to indulge this sort of playing.
However, avoid players pushing their creative responses so that other players' abilities get trumped by any old whacky idea.
For example, if a solution you devise results in this player easily volleying slingshots back into kobolds, while the mage is burning spell slots on Shield, the monk is feverishly deflecting missile, and the cleric is healing everybody who isn't holding a baseball bat, then this isn't going to be a fun little lark for the party as a whole, and they may start complaining about how unrealistic this is.
Reward the creative play the way the book dictates
Oh, and grant the player Inspiration, of course. This is going to be fun.
It would look like being a 3rd-level Monk (PHB p. 78):
Starting at 3rd level, you can use your reaction to deflect or catch the missile when you are hit by a ranged weapon attack…
It's just not something that's reasonable or feasible for someone to do without specific training — even top-performing baseball batters can't reliably react usefully to a ball that's aimed at them instead of into the box, even to get out of the way, let alone hit it.
RAW: rules for baseball are not included.
RAW that should inform the ruling(s) to come: the Monk has a class feature, Deflect Missiles, that accomplishes a lot of what you're talking about. It's a reaction, not a readied action, and it works on damage reduction. It's keyed to DEX, level, and a d10 roll.
So on one hand, you don't want to too-easily allow every character to do what one class has as a feature. On the other hand, a Readied action is a much steeper price to pay than a reaction, so this might not be too crazy.
Best advice: you should present the idea to your group as a purely-experimental trial-run. Barely-unearthed Arcana, if you will. Run a few mock combats, see what you like and what you don't. Set criteria beforehand, and evaluate as a group. One criterion should be, in my experience, whether this slows combat. And if you're giving characters a new way to avoid damage, it may be very good at doing that. My groups have tended to reject any houserule that slows combat--YMMV.
DEX seems like a reasonable component to include. Monks are specially training for this sort of shenanigan and get their level, so others shouldn't get something as "powerful" as level going into it.
Unless the character specifically has a feat or other trait that allows them to reflect projectiles, this shouldn't be possible. There is a common misconception that slings are weak, glorified toys. Many depictions of slings show the rocks being fired slower than a person could throw them, and they're depicted as a nuisance rather than a real weapon.
In actuality, a good hit from a sling is comparable in damage to a low caliber gunshot. These rocks (or if they're decently equipped, egg shaped hunks of lead) are travelling very fast. As others have said, deflecting a thrown ball is difficult, and balls aren't designed specifically to kill you. Deflecting a sling, even if you hit the projectile, just means that whatever part of your body you tried to deflect it with is the first part hit.
Is this allowed? What would it look like mechanically?
So, that sounds like fun, but is it realistic to play "baseball" with slingstones?
There are a lot of factors that might be reasonable about it, but there is one big problem: a baseball batter only needs to hit a ball pitched into a specific strike zone. The kobolds won't be aiming for the strike zone, they'll be aiming for him.
Below is detailed analysis comparing slingshot with "big league" baseball.
A major league baseball fastball goes about 42 m/s (95 MPH) or more. Slinging.org reports that leaden slingshot (that is, the high-quality kind, better than rocks or baked clay) can be thrown at a speed of up to 33 m/s (78 MPH) from a classical Egyptian sling - that's the speed of a major league changeup. So the slingshot is actually in the lower range of major league baseball pitch speed, a speed not too fast to be hit.
A typical slingshot weighs 40-160 g (1.4-5.6 oz), while a standard U.S. baseball must be within 142 and 148 g (5 and 5 1/4) ounces. So the weight range is also comparable. So the club could deflect the slingshot - it wouldn't be like trying to bat away a bowling ball.
A baseball is 72.64–74.68 mm (2 7/8 – 2 5/16 inches) in diameter while a large-end slingshot would be about 39 x 22 x 16mm (1 1/2 x 7/8 x 5/8 inches).
Since the slingstone is roughly half the size of the baseball, it would clearly be harder to hit.
It's time to finally bring up a very important factor. A baseball pitcher is (usually) not trying to hit the player with the ball. A baseball batter only needs to hit pitches in the strike zone while our slingstone batter would need to dodge out of the way of a well-aimed shot and then hit it.
Some of the stones would be coming in higher or lower than the strike zone where they would be more difficult to hit--such as at the target's head.
Any baseball fan could tell you that if a pitch, even an changeup, is coming at a batter's head, he tries to get out of the way of the pitch. He doesn't try to hit it.
If the slingshot is heading for the target (that is, the hit roll was successful) and the slinger is standing comparably as far away as a baseball pitcher (that is, around 60 feet), then any baseball fan can tell you that even a superstar cannot bat the incoming projectiles away.
Unless you are willing to consider your hero PC's to be greatly more athletic than major league baseball players who train to do a similar thing, batting away slingshot is not plausible.
If your campaign emphasizes a gritty realism, then there is no reason to extend the rules to make this feasible.
Applying what we know about baseball, an attempt to bat away a slingshot directed at you should have a very high DC.
As baseball pitchers rarely have trouble hitting a batter when they choose to, granting the attackers advantage on hitting the loon might be appropriate as well, as he is has stated he's just standing out there in the open with his club.
