One method that is easy to use and easy to overuse is the ellipsis.
"The solution to your problem is ... difficult to relate. Not so much for it's complexity but because I'm sure it will cause offense."
This method when used sparingly is effective, and when abused is very annoying.
An effective method is to use beats and gestures since they are two techniques to communicate a character's inner state during dialogue or action moments.
"Where should I start?" John's words flowed like logs on a meandering river. "It's not enough to say this is beginning. That misses the reason for why all this came to pass." He beamed broadly as he said this last bit. "And, that is the really clever bit."
"How can I make you understand," John said. The index finger of his right hand extended and punctuating the points he thought most salient, his words, not mine. John talked like that, using one word where twelve would do. "First, you put your left hand in. Then, you take it out." He mimed the actions as he spoke. "Then you turn yourself around. That is what it is all about."
Another method is to break up the dialog into individual lines. This gives the effect of pauses and breaks in fluent speech. It doesn't work well in this medium because of formatting issues. But the idea is to use the period since it signifies a full stop but use it on incomplete sentences. Our reading minds will recognize it as pauses in speech.
"I never thought that."
"I'd see the light."
"Of day again."