Rhodes Island is famous for its good weather, especially for the many days of sunshine, thats why it also have the nickname: "The Island of the Sun", and its also one reason the island is such a popular travel destination...
Following you will find some weather charts that will hopefully help you decide what is the best date for your visit.
Note that July and August are REALLY HOT months, and can be quite expensive too since the island has the largest number of visitors, this is the high season of Rhodes. May, June, September and October can be very nice months to visit Rhodes, especially September which is still hot but not very much..

If you wish to visit archaeological sites and do sightseeing, July and August could be very difficult, because some sites will require you to walk uphill at difficult places under the hot sun!!! Try June, September

If you wish to spend a relaxing holiday by the beach, Hot months are fine, but you need to make sure you wear a good uv protection cream since uv indexes are quite high during June, July and August, if you have very white / sensitive skin, I would suggest you avoid these months, or be very very careful with the protection against the sun. The water temperatures for swimming are very comfortable at August and September. Earlier months it can be a bit chilly.

WARNINGS / TIPS- Things to watch for your health.

In any case, you would want to get a room with A/C or at least fans, especially if you decide to stay at Lindos village (which can have very high temperatures day and night due to its location) or Symi Island. Dont risk it, otherwise you may spend sleepless nights, even the locals have hard times at evenings during July / August without A/C.

Kids should be very well protected with uv protection lotions with high index, no matter which season you visit Rhodes, and in addition with clothing even at the beach and while swimming, for extra protection. Hats are always necessary, as well as good sunglasses with uv protection.

In case of sunburns, which its most likely you will have at a small or larger degree, don't try to sleep it off without treatment, make sure you visit at least the pharmacy for some treatment (usually aloe vera lotions are quite good and natural), sleeping with a sunburn can be impossible, and at night its a bit hard to find a pharmacy open. If you think that getting in the water will protect you from sunburn, you are wrong, in most cases it will make things worse especially if you leave the water on you, the droplets are becoming lenses that concentrate the sunlight and may cause you a sunburn. Sitting under an umbrella will protect you but not 100%. The sun rays are reflected and get to you even under a shade. You can easily get a sunburn without protective lotions with high protection index.