Ah, it doesn't look brilliant but had a look at
http://www.wunderground.com/global/stations/16749.html which i normally find pretty reliable. They only have Sunday as a definite rainy day, other days they list as partly cloudy/scattered clouds with a chance of rain so if you are unlucky enough to see rain it'll probably be just a shower (fingers crossed).
Well Rhodes Town seems an obvious choice. Head into the New Town and check out some of the boutiques, it's amazing how a good spending spree can bring some sunshine into your life. You'll also find several umbrella shops in this part of town, which could be useful! There's The Palace of Grandmasters, Galleries and Museums in the Old Town which will give you something to do indoors, the Old Town too has plenty of shops to check out.
Failing that do what i do when you're on holiday and the rain is pouring... find a nice bar and drown your sorrows!