In the group I DM for, I like to give out birthday gifts to my players. The person I am making this for is a Beast Master Ranger that loves Alchemy. I was hoping to get some feed back and some ideas on how to balance this item for the game. I was hoping to give them a way to do magic damage, while storing his crafting supplies and potions. I have listed below the description of the magic item below. Thank you for your help in advance. 

Elemental Quiver of Holding

Each of the quiver's three compartments connects to an extradimensional space that allows the quiver to hold numerous items while never weighing more than 5 pounds. The shortest compartment can hold up to sixty arrows, bolts, or similar objects. The midsize compartment can hold perishable items while keeping them fresh and 10 potions without fear of them breaking. The longest compartment holds up to six long objects, such as bows, quarterstaffs, or spears.

You can draw any arrow from the small compartment as if from any quiver. Alternatively, as a bonus action the wielder may speak the command word, which infuses the arrows with the elements and becomes magical. 

Fire - You shoot a red hot arrow at a creature or object within range. Make a ranged melee attack against the target. On a hit, the target takes 1d8+Dex Fire damage. A flammable object hit by this arrow ignites if it isn't being worn or carried. (The damage increases by 1d8 when you reach 5th level (2d8), 11th level (3d8), and 17th level (4d8).)

Ice - You shoot a frozen arrow at a creature or object within range. Make a ranged melee attack against the target. On a hit, it takes 1d8+Dex Cold damage, and must success on a DC14 Strength save or its speed is reduced by 10 feet until the start of your next turn. (The damage increases by 1d8 when you reach 5th level (2d8), 11th level (3d8), and 17th level (4d8).)

Lighting - You shoot a blue arrow at a creature or object within range. Make a ranged melee attack against the target. On a hit, the target takes 1d8+Dex Lighting damage. (The damage increases by 1d8 when you reach 5th level (2d8), 11th level (3d8), and 17th level (4d8).)

Air- You shoot a clear arrow at a creature or object within range. Make a range melee attack against the target. On a hit, the target must make a DC12 Strength save or get push 10 feet back. If it is an inanimate object it auto fails getting thrown 10 feet. (The save increases when you gain levels. When you reach 5th level (DC14), 11th (DC16).)

First, what's a ranged melee attack? I have an idea based on 3.5, but I'm not familiar with this in 5th edition.

I don't know what power level your game is, but the arrows seem a bit much. Just being able to swap elements by itself is really good since you can overcome a lot of Resistance and potentially have the right Vulnerability when you need it.

If you don't want to limit how often you can pull an elemental arrow between long rests, then I would suggest limiting the secondary effect of each element to maybe once or twice per long rest (lightning doesn't have one, btw).

1. Fix the typo of "ranged melee attack"
2. Require attunement to gain the magical effects of this item
3. Depending on what tier of play the item is for, the quiver should have a limit use. I personally think it should have something like 5-6 uses of their choice and regain at the end of a long rest or regain x number of charge at dawn.  Though Save_Versus's suggestion of 2-3 per element, is another fine change too.
4. When it comes to the effects of each element, I find one thing obviously out of tune. That would be that, cold is by far the best option (unless the target is resistant or immune)  because the bonus damage is the same as fire and lightning but also has a chance to add a secondary effect. I would either suggest to add secondary effects to fire and lightning (which is my preference), or lower the damage of the cold so if they wanted to secondary effect they aren't gaining as much damage.
5. For the Air arrow, I would also add the effect of "and is also knocked prone" simply because it doesn't give a bonus damage and if this is going to have charges then it need a little more of a reason to pick it. 

I agree that these seem a bit too powerful... honestly just adding +1d4 elemental damage plus a status element to the damage of an attack would be pretty boss, so giving it a number of charges per day should still be in effect. 6 seems like a healthy amount.

As for suggestions for the different element types... fire should probably ignite the opponent so they take some extra damage their next turn (probably just another d4), the effect for Ice already seems good, Lightning I think would force the opponent to not be able to take a reaction this round (similar to shocking grasp), and i agree that the air arrow should go ahead and just knock the opponent prone.

Just spitballing here, but I'm thinking of these arrows as "the right tools for the job" sort of deal, so perhaps focus on their utility?

I'd keep the damage boost per level, but remove it from the arrow descriptions, just make it a default ability. I'd also remove the elemental damage types for the reasons I stated above.

Instead of Fire, I'd go with Burning Arrow and have it do persistent damage until your target spends their action to remove the arrow. (For high HP enemies)

Instead of Ice, perhaps the secondary effect turns the area into difficult terrain? (For fast enemies)

Instead of Lightning or Air, I'd go with a Bolt Arrow that's so fast that they have to make a Dex Save instead of a regular attack roll. (For high AC enemies)

If you want 4 arrows, maybe an earth/nature arrow, but I can't think of what'd it do.

Earth arrow sounds like a good excuse to knock an enemy prone. Like you shoot the arrow at their feet and it creates a localized earth tremor and they have to make a DEX save or be knocked prone.

If the OP already has an effect that affects movement, you don't really want to double dip. Reducing movement by 10 ft and knocking prone (0 ft) are both in the same wheelhouse.

Also, Prone isn't that great in 5e. If we're within reach of each other and I knock you prone, you just stand up on your turn and strike me and it only cost you some of your movement. I never get to capitalize on you being knocked down.

In keeping with the utility idea, the only other thing I could think of was like a Boulder Arrow that dazes or stuns the enemy, but that's getting into "Green Arrow Boxing Glove Arrow" territory.

Ah good point, that would pretty much make the Ice Arrow redundant... hmm, maybe it's something that's only useful against airborne enemies. Like it temporarily increases their weight and they have to make a strength saving throw or drop... 10 feet? 30 Feet? I'm not sure what would be most useful without being too broken. But either way, this would be a blessing for all the melee characters chucking javelins and hand axes at anything that starts flying away.

So.  Different idea completely.

The elemental arrows are a cool idea, I have a Scintillating Quiver myself I put together for elemental archers, but this is supposed to be for an alchemist who loves crafting stuff and is primarily intended to store the alchemist's arrows and alchemical supplies both.  The elemental arrow feature kinda comes out of nowhere given the overall theming of the item.

What if, instead, the quiver could apply the ranger's concoctions to their arrows as a bonus?  Something to the effect of...:

"Apothecary's Quiver

This magical quiver has three compartments, each of which connects to an extradimensional space that can contain many small items.  The quiver itself always weighs five pounds, regardless of what it contains.  The smallest compartment can store up to sixty arrows, bolts, or similar items.  The medium compartment can hold up to ten pounds of perishable objects or ingredients, which remain cold and fresh indefinitely while stored within the quiver, as well as up to ten potions, poisons, salves, or similar items without fear of them breaking or spoiling.  The largest compartment can contain a single set of field alchemist's tools.

You can draw an arrow from this quiver freely, as per any normal quiver.  Additionally, as a bonus action the wielder may command the quiver to apply the effects of any potion stored in the medium compartment to any piece of ammunition stored in the smaller pocket, consuming the potion in the process.  This arrow is produced the next time the wielder draws an arrow from the quiver.  If a beneficial potion is applied, the wielder may choose to deal no damage with their next attack this turn; if a poison or other harmful potion is applied, the arrow functions normally.  Any creature struck with an arrow so treated also gains the benefits or suffers the harmful effects of whatever potion was applied to the arrow that struck them."

This way, the player can easily shoot their alchemy however far away it needs to go, delivering healing potions to anywhere within bowshot or gaining the ability to easily utilize any poisons they may create.  A smokebomb-type potion turns into a smoking arrow, while an exploding potion turns into an explosive arrow.  The possibilities would be endless, for a properly creative alchemist - especially since the quiver also contains a basic set of tools for him to replenish his stock in the field.

As much fun as I've had theorizing ideas for how to make the originally designed quiver balanced and interesting, I have to say that I like this new idea a lot more thematically. It does a lot more to make this a gift very specifically to reward an alchemist player.

I agree. Thank you everyone for all your inputs 

If you want to keep the theme of the "elemental" part of the quiver, then what you could do is simply turn the arrows into magical damage of the chosen type. No extra damage, just turns piercing into fire damage, etc.