Likely Uncommon
Given the word "Luck", it at first made me think of the Lucky feat (PHB, p. 167):
You have 3 luck points. Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20. You can choose to spend one of your luck points after you roll the die, but before the outcome is determined. You choose which of the d20s is used for the attack roll, ability check, or saving throw.
However, on reflection, as it's currently worded, it's actually more similar to a fighter's Indomitable feature (PHB, p. 72):
... you can reroll a saving throw that you fail. If you do so, you must use the new roll, and you can’t use this feature again until you finish a long rest.
Of course, Indomitable is only for saving throws, whereas your item is for attack rolls and ability check as well, which then makes it more like the Lucky feat, but unlike the Lucky feat, you must use the new roll and it can only be used once, making it more like Indomitable again. In other words, it's somewhere between the two.
The reason I say this is Uncommon is because, unlike both Lucky and Indomitable, you have to anticipate its use beforehand, because you can only use it within an hour of activating it. Lucky and Indomitable can be used reactively with no preparation or forethought ahead of time. For this reason, I think it's niche enough that uncommon sound about right.
In Explorer's Guide to Wildemount (p. 186), there's a spell called fortune's favor (thanks @MivaScott), which provides a similar benefit to the above items:
You impart latent luck to yourself or one willing creature you can see within range. When the chosen creature makes an attack roll, an ability check, or a saving throw before the spell ends, it can dismiss this spell on itself to roll an additional d20 and choose which of the d20s to use. Alternatively, when an attack roll is made against the chosen creature, it can dismiss this spell on itself to roll a d20 and choose which of the d20s to use, the one it rolled or the one the attacker rolled.
If the original d20 roll has advantage or disadvantage, the creature rolls the additional d20 after advantage or disadvantage has been applied to the original roll.
At Higher Levels. When you cast this spell using a spell slot of 3rd level or higher, you can target one additional creature for each slot level above 2nd.
This is a 2nd level spell that lasts for one hour, similar to your magic item, although it can be cast on anyone nearby, whereas your item only affects you. That said, it implies that such an ability is equivalent to a second level spell.
As an example of another magic item that casts a spell, the Bowl of Commanding Water Elementals, which is Rare, effectively lets you cast conjure elemental, a 5th level spell (although it forces you to have a specific elemental). A better example perhaps is one of the various wands that let you actually cast spells, 2nd level spells in particular; the Wand of Web is an Uncommon magic that let's you cast the web spell (a 2nd level spell) seven times. So that also sets a precedent for a magic item being able to cast a 2nd level spell being Uncommon.
Another point of comparison is the Potion of Possibility from Explorer's Guide to Wildemount (p. 268), which appears to be a stronger version of your item:
When you drink this clear potion, you gain two Fragments of Possibility, each of which looks like a Tiny, grayish bead of energy that follows you around, staying within 1 foot of you at all times. Each fragment lasts for 8 hours or until used.
When you make an attack roll, an ability check, or a saving throw, you can expend your fragment to roll an additional d20 and choose which of the d20s to use. Alternatively, when an attack roll is made against you, you can expend your fragment to roll a d20 and choose which of the d20s to use, the one you rolled or the one the attacker rolled.
If the original d20 roll has advantage or disadvantage, you roll your d20 after advantage or disadvantage has been applied to the original roll.
This item gives you two uses rather than one, which must be used within 8 hours rather than 1, and works more like the Lucky feat than yours (which is somewhere between Lucky and Indomitable). Given that this item is very rare and can do a lot more than yours, I'm still happy with my assessment of Uncommon.
That said, the potion is a one time use item, whereas your item is not. Therefore, another comparison could be made between your item and the Luxon Beacon, from the same page of Explorer's Guide to Wildemount:
Fragment of Possibility. A creature that touches the beacon and concentrates for 1 minute receives a Fragment of Possibility, which looks like a Tiny, grayish bead of energy that follows the creature around, staying within 1 foot of it at all times. The fragment lasts for 8 hours or until used. Once the beacon grants a Fragment of Possibility, it can't grant another until the next dawn. A creature with a Fragment of Possibility from a Luxon beacon can't gain another Fragment of Possibility from any source.
When a creature with a Fragment of Possibility makes an attack roll, an ability check, or a saving throw, it can expend its fragment to roll an additional d20 and choose which of the d20s to use. Alternatively, when an attack roll is made against the creature, it can expend its fragment to roll a d20 and choose which of the d20s to use, the one it rolled or the one the attacker rolled.
If the original d20 roll has advantage or disadvantage, the creature rolls its d20 after advantage or disadvantage has been applied to the original roll.
This allows one per day, although it must still be used ahead of time as it only lasts 8 hours, but this is still degrees more powerful than your proposed item. The Luxon Beacon is given the rarity of Legendary, so it better be degrees more powerful if I'm claiming yours is Uncommon.
Another comparison, from the DMG (p. 179) this time (thanks @Medix2), the Luck Blade, which allows a reroll once per day:
Luck. If the sword is on your person, you can call on its luck (no action required) to reroll one attack roll, ability check, or saving throw you dislike. You must use the second roll. This property can't be used again until the next dawn.
This does force you to use the new roll, like your ability, but it can also be used reactively with no forethought, like the Lucky feat. Unfortunately, since it can also cast wish, its rated as Legendary and I'm not sure what it would be without being able to cast wish, so that doesn't really give us much to go on...
Finally, as one more point of comparison (again, thanks @Medix2), the Chronurgy wizard archetype from Explorer's Guide to Wildemount (p. 184) has a class feature that they get at 2nd level called Chronal Shift:
You can magically exert limited control over the flow of time around a creature. As a reaction, after you or a creature you can see within 30 feet of you makes an attack roll, an ability check, or a saving throw, you can force the creature to reroll. You make this decision after you see whether the roll succeeds or fails. The target must use the result of the second roll.
You can use this ability twice, and you regain any expended uses when you finish a long rest.
So this is twice per long rest, rather than once, and can be used reactively, rather than needed to prepare it ahead of time. It does force you to use the second roll, though, but it can extend to anyone nearby, so that's pretty strong. Since your item is weaker than a 2nd level class feature, I think I'm still happy to call this an Uncommon item.
