It's not game-breaking, but it's a bit much.

So the first ting to note is that invocations vary wildly in power. Some are absolute game-changers. Others are easily replaced with minor magical items, or allow one-per-day casting of spells that weren't all that good to begin with at the cost of a pact magic slot. This invocation won't be the best, and it won't be the worst. The only questions we can really even try to answer are "Is this so good that it notably increases the power of the warlock class overall?", "Is this so good that all or nearly all warlocks would want to take it?", and "Is this so bad that taking it would be an obvious mistake in most or all cases?"

What does it give?

• You save one "spells known" slot. Those are pretty valuable.
• You effectively get to reassign your hex target at will with a bonus action (by dropping concentration and reapplying). Standard hex requires you to reduce your initial target to 0 HP before you can reassign
• Failing a concentration check becomes much less big of a deal.
• It allows you to dance back and forth between Hex and other concentration spells much more easily, effectively reducing the cost of a number of useful/potent spells (at 5th level, Invisibility and Suggestion stand out especially, but there are certainly others.)

This is a strong invocation. I don't think it's strong enough to significantly increase the overall power level of the Warlock, but it's a definite bump, and it's strong enough that I'd probably try to find a way to fit it into basically any Warlock I built. It also makes Hex tactically less interesting. The tradeoff of when to drop Hex for other concentration spells, and the concern of where to go to find something hapless to murder to bring it back up again are interesting questions. The limitation of having to pick a specific target and not being able to reassign until you finish them off is also interesting. This simplifies that all away to nothing. Then it also saves you a spell known on top of that.

So, if you have a Warlock who doesn't want to deal with those sorts of questions, and you want to make life easier on them, making this available as an invocation will do that. It likely won't make them overpowered unless they were already one of the more powergamer players at the table already. If they are, they don't need the help. If you're trying to find a balanced form, though, I wouldn't describe this as balanced.

For something that's more balanced, I'd suggest a once-per-short-rest ability that let you cast Hex at the current level of your Pact Magic, and required that Hex be on your Spells Known list. Basically, it gives you an extra Pact Magic slot that can only be used on Hex. This still gives you your ability to use non-combat concentration spells and combat-opener spells and bring Hex up afterwards, and/or lets you reassign without dropping the foe or recover from lost concentration once, but it's more limited - it offers a bit of flexibility on those things, rather than simply removing the restraints altogether. Having it be at your Pact Magic level rather than at level 1 just means that you don't have to worry about swapping back to the "good" version. This invocation is clearly intended to simplify your life, and that's extra fiddly complexity. That's an invocation that... well, honestly, I'd probably still take it eventually, but it would be a lot lower on my priority list.

It's not inherently unbalanced, but it breaks theme

In essence, you're moving the Sorcerer's ability to turn spell slots into other spell slots to the Warlock, with the added bonus of being able to do it in one turn. (A sorcerer needs to turn a spell slot into spell points first, and then turn those spell points into new spell slots.)

In itself, this isn't game breaking, you get a maximum of 20 spell levels worth of slots, as opposed to a Sorcerer's 100+ at level 20, but you recover those spell levels at a short rest, and a Sorcerer regains them on a long rest.

However, it does come with a problem:

You are breaking theme

D&D tries to give all the classes at least some defining features, and you're essentially taking one of the cool things Sorcerer has going on and giving it to the Warlock. Is this a problem? It depends. If you have a Sorcerer in your game too, they may very well feel that they're getting the short end of the stick. Warlocks get plenty of Invocations, so spending one to get this perk isn't that big a deal, and the result is that this Warlock is now just as good at spamming low level spells as a Sorcerer is.

In my experience, Sorcerers are already less popular than Warlocks are at my table, so if I implemented this Invocation, I'd be almost 100% certain nobody would play a Sorcerer anymore.

It's overly complex

Your description is very wordy, and it's kind of hard to figure out what exactly the ability does. In addition, Warlocks regain their spell slots on a short rest, so the whole 'splinters get lost' part can just be shortened to something like 'when you recover your spell slots, unused splinters disappear'.

Personally, I'd do away with the whole splinter naming theme and just call them spell slots, it's a lot less confusing.

It might also become mandatory

Comparing your new Invocation to the other available Invocations, it is a lot stronger than many of the other ones you could pick. While it doesn't automatically break game balance, I feel like anybody who plays a Warlock would be guaranteed to pick this Invocation up as well, as it allows you to cast way more spells in a day, and you are flexible in what those spells are.

In itself this isn't a huge problem, Agonizing Blast tends to fill a similar position of 'every Warlock has this Invocation', but it will result in your Warlocks feeling more similar.

Personal opinion

I wouldn't ever add this to my games. I feel that it's stepping too much on a Sorcerer's toes, it breaks down the Warlock's overall theme and it feels a bit too strong and complex for a single Invocation, but I doubt your game will crash and burn if you use it. It's strong, but not gamebreakingly so.