Hoyer is currently down as the team's starter for the 2014 season but suffered a spate of injury problems last year and is going into the new campaign off the back of knee surgery.
Alex Tanney is the only other quarterback currently on the Browns' roster having joined the team late last season as cover for the growing injury crisis.
Young has not played in the NFL since 2011, when he featured for the Philadelphia Eagles, although he did spend the 2013 pre-season with the Green Bay Packers and Browns general manager Ray Farmer is keen to give him a chance to work his way back into the league.
"He's had success in the National Football League. We wanted to take the opportunity to get the chance to see where he was," Farmer told The AP.
"We'd been kind of following him through the off-season. He'd been doing some things that kind of caught our eye and made us kind of take note."
Thigpen made four appearances for the Buffalo Bills in 2012 and has 21 touchdowns and 18 interceptions in his 29 NFL games overall.