You've already given part of the answer: the suffix -ito (and occasionally other diminutive suffixes such as -illo) can be added to words as a sign of affection or friendliness. When used in such a way, the suffix doesn't really affect the meaning of the word itself, and it certainly doesn't refer to size, but it indicates the attitude of the speaker toward the listener.
As you probably know, diminutive endings are used most commonly with nouns. But they are also used fairly frequently with adjectives, and they in fact can be used with adverbs and even (as in this case) interjections. (One of the most common examples of a diminutive ending for an adverb is ahorita, a diminutive (more common in Latin America) meaning "right now," taken from ahora, meaning "now.")
In the case of the example you mention, the -ito suffix has been added to an interjection chau a friendly way of saying goodbye. (Often spelled ciao and sometimes chao, the word comes from Italian.) So by saying chauito, the person was just being especially friendly and casual. In English, we might say something similar with "bye-bye" or "bye now." Those terms you might use with a good friend, but you probably wouldn't use them if you had just met an authority figure you're trying to impress.
Along the same line, you'll sometimes hear the term adiosito as a way of saying good-bye. And holita is sometimes heard as a friendly way of saying hi.
A common example of how the diminutive might be used to indicate affection or friendliness rather than to change the meaning of a word is in the making of requests. In a restaurant, for example, you might ask for un cafecito, por favor. You're not asking for a little cup of coffee in that case, just being friendly and sounding less demanding. There's not really a direct translation in English, but you might express something similar by asking for "a nice cup of coffee, please." In that case you're not really describing or passing judgment on the cup or coffee, just being friendly. And so it often is when you use diminutive forms of words when indicating something other than size.