On different trips, I've stayed in the Negresco (pre-renovation, ugh!), Hotel du Cap, Chevre d'Or, a famous restaurant with rooms in Cagnes les Haut that has since gone out of business, and the Chateau St Martin.

Hotel du Cap is tough to get into in high season and tends to favor its regulars, some of whom are the international young newly-rich and beautiful jet set crowd.

Be aware that Eze forbids cars, so that arrival and departure are inconvenient, IMO Eze would not be a convenient base for this reason. It's a cute historic village for a day or so with nice views above the sea, although it can seem touristic by day. I remember my room in Chevre d'Or as much more quaint and cozy than luxurious but YMMV. I was not impressed by their food. There's also a Chateau d'Eza which I have not tried.

It's not the same category as Hotel du Cap (although the owners are the same), but I really liked Chateau St Martin and felt very welcomed; it has a reputation for treating AmEx FHR clients very well and I would agree. I liked being greeted by the manager on duty and having the front desk manager volunteer information about my two-category (above and beyond) upgrade (to a deluxe junior suite in the tower with a view) at check in. Good food too (IIRC, Michelin two stars) in a pleasant setting with great views, plus a less formal outdoor restaurant near the pool featuring grilled things. The pool is merely OK; not heated and not particularly pretty or stunning, at some distance from the main building but with no views.