The power-level is appropriate and not game-breaking.
This watch specifies that the user has an action that they may use to either perform an ability check or to gain advantage on a Dexterity saving throw (and not any other actions such as making an attack or casting a spell). By specifying the action thusly, you rule out a lot of exploitable situations - unless the table uses a considerate amount of house rules that interfere. For a 7th-level character, even on a one-hour cycle, the power level is well below other major magic items that they might access.
The game effects behave like a significantly nerfed version of a Haste spell without most of its usability - a third level spell would be appropriate for that rarity (for reference compare to Magic Item Table G), and while a one-hour cycle lets you use the item an unusual amount of times, the niche benefit of performing an ability check is a ribbon in many situations that empowers the player to use actions they often don't use in a tight action-economy. The advantage for Dexterity saving throws is more conditional and weaker than that of many major rare items, but also has a broader use than most of them (in particular compared to flat advantage against a broad category such as granted by the Mantle of Spell Resistance or a narrow category such as granted by the Dragonguard).
The rarity is unusual - complicated, and you might want to adjust it. But does it matter?
Major rare items with entirely different but comparable effects tend to have charges that restore at dawn/dusk, a short or long rest. Wands usually hold more charges than rings and have a chance to crumble to dust. Essentially this magical watch has a highly unusual amount of charges between rest circles. The amount is more akin to major very rare magic items like the ring of regeneration that restores 1d6hp every 10 minutes.
In its current iteration, its activation method makes it more akin to major very rare, wondrous item, item behaviour. Major rare items tend to have set conditions compared to very rare major items. Generally, the rarity categorisation is a rather vague concept, and the categories and the implied availability should service what happens at your table. The DMG uses the legendary major item ring of invisibility as a narrative storytelling tool as an example - see the Rarity section in the DMG.
Ask yourself why the item rarity matters in this particular situation and pay attention to unnecessary restrictions that might hinder enjoyment at your table. Rarity is a pacing and availability tool - that is the primary concept underlying handing out magic items to your players. See the DMG chapter seven and XGtE chapter two, which provide guidelines on how many magic items a party should have by each level. We also have Q&As about the power level and rarity and the vagueness of rarity when looking at the rarity of mundane items.
I would suggest changing the verbiage slightly and adding attunement to make it more in line with a major rare item:
Wondrous Item, rare (requires attunement)
This silver pocket watch's inscription on the inside of the cover
reads: "Take the time to think". While wearing it, you may activate it
as a reaction to think and perceive at an accelerated speed for a few
subjective seconds while time around you seems to move slower. You may
take a single action to make an ability check that does not require
moving your body, such as Wisdom (Perception) or a roll to recall
facts about a subject. Alternatively, you may activate the watch in
response to one effect that requires you to make a Dexterity saving
throw, you grant yourself advantage on that roll. Once activated, you
cannot activate the watch again for one hour.
This formulation adds precision to the number of effects and its cooldown and uses accessible language - the flavouring is a flourish that doesn't change the effect. I believe it makes it more comprehensible.
Additionally, one hour is difficult to track for the GM, and you equip the player with something that not only tells the time but integrates it as important due to their fun magic item. Making time more important might change dynamics in a way that gives you less flexibility and creates more work at the table.