To Mention, To Say, To Eat, To Drink
transitive verb, expression
You've already learned that 口 means "mouth" and here it's followed by the particle に, which pinpoints a location. The verb する means "to do" so you're doing something to your mouth! Logically enough, this expression is used to describe lots of actions that you do with your mouth. It's often used to mean to mention or to say, but with an added descriptive nuance that you've said something accidentally or unexpectedly, maybe something you shouldn't have said! 🙊 It can also mean other things done with the mouth, like to eat and to drink!
くちにする
In this expression, 口 is a standalone vocabulary word, so it uses the same reading that you learned when you learned the vocabulary.
女王は、よく「そうね」と口にする。
The queen often says, "you're right."
その名は口にするな。
Don't say that name.
ウィスキーを口にしたら、よっぱらってしまった。
I got drunk off a little whisky.