The 20-year-old is a free agent after completing a short-term contract with League Two Chesterfield last season.
Brindley, who began his career at Norwich, rejected the chance to join a Championship club to join the Millers, who begin life back in League One next season.
"We met Richard and his representatives over the past few days and the bottom line was he was either going to sign for us or a Championship club," Evans said.
"I think he felt it would be a real fight to be involved in the first team at the Championship club, where as we were talking about him playing in the first team now.
"He is a young man that we first watched when he was at Norwich and we received a ringing endorsement of his ability from Paul Lambert.
"We are very excited at having him on board for next season."