Provided your connection is ok with being mentioned/name dropped, I'd name your connection just after you introduce yourself in your cover letter. The name/connection must be early on in the letter to grab the reader's attention, otherwise the person reviewing it is apt to skim it and miss it entirely, which does you no good.
So after you say, "My name is so-and-so, a rising 2L at school X," perhaps put something like, "Mr. Name Drop has shared a lot of information with me about the firm; I am especially interested in X practice area" or something like that. Then go on to what you can bring to the firm in your second paragraph.
Using connections is so much easier outside of OCI, when you have the easy introduction of, "so and so recommended I contact you."