Monaco is different enough from the rest of the Riviera to deserve a proper visit, but not large enough to require a hotel change by default. For a first trip based in Nice, a return day trip works extremely well.
Come for the day if you want to see Monaco. Stay overnight only if you want to live Monaco.
Monaco for the day
A day gives you time to combine the Rock, the harbour, Monte-Carlo and one major sight such as the Oceanographic Museum without turning the visit into a competitive event. The territory is small but vertical: horizontal distances have an unfortunate habit of forgetting the stairs.
The obvious trap is reducing Monaco to the Casino, cars and shop windows. Entertaining for twenty minutes. A full day can aim slightly higher.
When one night becomes interesting
Stay in Monaco if you have an evening restaurant or event, if the hotel is itself part of the holiday, or if you want the place before and after the day-trip crowds. Then the premium starts buying more than a postcode.
If the plan is basically “see Monaco, continue to Menton”, keep the hotel in Nice and spend the difference on something you will remember more clearly.
What actually deserves time
The Rock and the Oceanographic Museum show a different side of the territory from Monte-Carlo. The harbour connects the two. The rest depends on your tastes: architecture, gardens, shopping, grand hotels or the simple pleasure of watching a place that decided long ago modesty was not part of the brand guidelines.
Do not visit Monaco purely to look at richer people.
That activity is freely available online and much kinder to the calves.
Transport: the train removes most of the drama
From Nice, Monaco works very well without a car. The station also removes one of the Principality's less glamorous experiences: following signs through tunnels while trying to remember which underground car park you entered.
For a first Riviera trip, a car adds little to this particular day. Save it for places where it actually changes access.
Hotels: our shortlist is intentionally theatrical
We currently have two very high-end references in or just above Monaco: the Hôtel de Paris Monte-Carlo and The Maybourne Riviera. That reflects what Monaco does exceptionally well. It does not yet solve every budget.
Monaco + Menton on the same day?
Yes, if you are willing to edit. Monaco in the morning and Menton later works logically along the coastal line. But if you want several museums, a long lunch and gardens, split the days.
Menton is a useful counterpoint: softer, less demonstrative and much easier to leave unperformed.