What don't your characters say aloud?
This is sometimes just as important a question as what they do say, and how they say it. If you find yourself asking questions about your dialogue, ask yourself one more - ask about what these people don't say, and whether they leave blanks in what they say. When we speak, our silences can be as meaningful as our words.
Just talking about silence may not make my meaning clear, however. Let me give some examples to illustrate some of the things that you can do with it. There are really (at least) two kinds of silence: someone not saying anything, and someone internalizing a reaction. Keep in mind that not answering, or keeping silence when someone else expects speech, is an active and not a neutral choice. In conversation, the default is to respond. When we don't answer, it means something.
If we call out to someone, and receive no response, it probably means they haven't heard us, and we should call again. This is the principle behind the ringing of the telephone. It calls, and gives us a chance to respond in the silence before it calls again. This is why I always say to my kids, "Geez, guys, give me a chance to reply before you start saying Mamma over and over!"
If we ask someone a question, and receive no response, it generally will be interpreted not as the inability to hear, but as a refusal to reply. The asker's subsequent reaction will vary according to how badly they take this perceived refusal.
If you're reading along in a conversation, and one of the speakers comes out with a reply that seems to have no bearing on what came before, it may appear to readers to be an error on the author's part, even if that was what you intended. However, if you're providing the internalized thoughts of your character, you can use what your character was thinking to bridge the gap - so that the character's next line will make sense to your reader, but not necessarily to the person the character was speaking to.
Silence and avoidance are good tools to set up mystery as well. I remember in Richard Adams' classic Watership Down, the heroes travel through a strange warren of rabbits who are stringent in their avoidance of any question that begins with "where..." The simple fact that these individuals would go so far out of their way socially to avoid "where"questions speaks volumes about the fact that something is wrong with this warren - which later turns out to be a place where the rabbits are fed by humans so that they can then be snared and eaten.
I used a different kind of avoidance strategy in my forthcoming story, "At Cross Purposes." The alien species in this story, the Cochee-coco, operates in pairs who are constantly together. I toyed with the idea of having them never use singular first person pronouns - "I," "my" etc. But it was too extreme, and indeed, very difficult to read and understand. So I changed my mind slightly, and decided to use a more subtle avoidance strategy - one in which the default value of self was the pair, and all actions and intentions were imputed to the pair, except in cases where one member or the other of the pair needed to be specified for some reason. So it wasn't that the Cochee-coco couldn't say "I," but they avoided it. The use of "I" was marked instead of being the natural default.
Certain societies (like the US) expect lots of talk to be made, and others (like Japan) expect their speakers to be able to guess more about what the other person is thinking and feeling. I come from a tradition of much talking (as my friends will testify), and so a good many of my characters have an easy time with words. However, the Imbati servant caste from my Varin world is a group whose primary role is that of keeping secrets and passing information when needed, so silence and inference have a very important job among them. I find it really wonderful and fun to write a conversation between two Imbati and to see how little they actually say aloud. It makes a chapter containing such a conversation much shorter! Here's an example of a girl, Kiti, greeting her boyfriend Aloran after a job interview, and the two of them deciding to speak at his bunk.
In the welcome light of the dormitory, Kiti jogged toward him between the long rows of steel bunk-beds. Perhaps reading his face, she didn't greet him with a smile. Instead she set her shoulder beside his and walked alongside him.
"Bunk," Aloran said.
He knows what she wants; she's guessed he's troubled, but sticks by him because she really wants to know what happened. All they need is to decide where to speak privately - one word.
I hope this gives you some ideas about how to have fun with silence.