We rented a villa in 2011, a bit to the east of where you're looking (Nice-Menton area). Did all my searching on either VRBO or HomeAway. Wound up in Beaulieu-sur-Mer with a nice little place perched on a hill overlooking the Med. It's a lovely town. There were six of us and we fitted in no problem. No pool, but the beach was only a short walk away. 600 euros/week.

This year, we're renting again. There will be 9 or 10 of us, so I had to find a different place. Settled on La Turbie, just above Monaco. Room for up to 12, heated pool, plenty of parking, etc. 800 euros/week.

The area is well situated for sightseeing locally, and the bus/train/road situation is such that getting anywhere is fairly cheap and convenient. The ambience can't be beat, to say nothing about the weather.