Check for 'Mougins'. It's a pretty nice hilly village (Indeed, all are hilly villages/towns when you get into the interior parts) also known as "Picasso's Village".
Other options include 'Grasse' (The world's perfume capital), 'Vence', 'Antibes', et al. If you plan well, you can probably find some cheap options.
Here's a link for you, if you wanna know more about 'French Riviera' :
French RivieraEDIT: Are you gonna use the public transport or your own vehicle, when staying around those places?