I think speech idiosyncrasies are a part of distinguishing characters from each other, but they are, more than anything, stylistic, writerly flourishes. Conan Doyle didn't define Sherlock Holmes's voice by excessive use of stock phrases. Sure, Holmes utters some choice expressions like, "You know my methods", but those are few and ornamental. What defines Holmes is the hyper-rational nature of his speech patter, which is a direct representation of his intellect: of his character. He speaks the way he does because he needs to speak that way because he needs to think that way to solve crimes with efficiency. It's a matter of necessity owing to Conan Doyle's understanding of his character's wants and needs.
I think it boils down to that awareness of necessity. That is what creates unique characters and distinguishable dialogue.
To understand those root desires and motivations, ask yourself these questions:
- What does my character want/need?
- What is my character willing to do to get it?
Variations on those questions, such as can be fleshed out using the very useful exercises suggested in other answers, will help you develop the answers to those two primary questions.
Once you know what your character truly wants, you will know how to phrase dialogue to communicate that want, effectively creating a new manner of speech distinct from your own. As an example: how does a 5-year-old respond to the question "Where are the cookies?" Well, that depends. Does the kid want to avoid punishment? Does she want to put the blame on her big brother who is mean to her? Does she, for some asinine reason, feel compelled to tell the truth about eating the cookies? You are less likely to sound like yourself in answering the cookie question if you understand what specifically your character wants to accomplish with her speech. This may sound simplistic, but the ardor comes in letting the question of necessity guide every dialogue passage.
Characters, unlike real people, speak for a reason. As readers we expect this. We expect to glean personality traits through dialogue. As a writer, understanding that readers are searching for this makes our job easier. We don't have to convince our reader to analyze our character in the same way that we might have to convince a love interest to take a fancy for us. We have only to understand a character and to express their desires in speech and the reader will do the rest.
NB: Sherlock Holmes never once utters the phrase "Elementary, my dear Watson". The closest he comes is in The Crooked Man where he responds to one of Watson's congratulations with, simply, "Elementary". FWIW.