what kind of name is it called when it is a play on words... such as "Sonny Rains?" Would the name just be called a pun?

Yes. Pun describes this kind of play on words.

yup...that would be a good name for that.

http://dictionary.reference.com/browse/pun

Yes, the name "Sonny Rains" would be considered a pun. A pun is a type of wordplay that exploits multiple meanings of a word, or words that sound similar but have different meanings, to create a humorous or clever effect. In this case, the name "Sonny Rains" is a pun because it sounds similar to the phrase "sunny rains," which is a contradictory combination of weather conditions (sunny and rainy).