The 35-year-old will take over, following the sacking of Aidy Boothroyd.
Frank Lampard Snr will also join Watford as a football consultant.
Rodgers said: “Frank is someone I have known for a little while now and our relationship came through young Frank, who I worked with at Chelsea.
“Talking to the man, and being in his company, I was always very interested in what he had to say and always came away having learned something. He has some great experience.
“As a young manager and coach, to me it was very, very important to have that someone alongside me. For me, at this moment in my career, I can think of no better man.
“He is a football man, he is an honest man and he is a loyal man. When you are a young manager those are the key qualities you want in someone who is your mentor really.”
