Stewart Jnr has been on the board of directors for the last four years and will work alongside his father as the Millers prepare to move back into the town to the New York Stadium.
"The new stadium is creating new opportunities for the football club, and to maximise this wonderful opportunity we need to be fully prepared for what will be an exciting new era for Rotherham United," said Tony Stewart.
"Behind the scenes Richard has played a vital role on the operational side of the football club and has been heavily involved in the development of the new stadium.
"The new stadium is guaranteed to move the club forward so it is important that I can continue to share the responsibility of key decision-making, and that is where Richard and other members of the board will be a factor."
Stewart Jnr added: "It is an honour to be handed this new responsibility at such an exciting time in Rotherham United's history. Having worked alongside my father successfully for the past 18 years, I am excited about being handed this extra responsibility, and I cannot wait to get started to help bring about success for Rotherham United."