Despite netting the winner in Monday's 3-2 victory at Blackburn Rovers, the striker has only scored nine goals in 43 appearances for the Reds since his £35m move in January 2011.
"I probably would start with Carroll [against Everton] because it always gives you an outlet and we know what he can be capable of on the day," Hamann told talkSPORT.
"If he doesn't come up with a big performance tomorrow I think you have to give up on him because I think people have been watching [his performances] for 15, 16 months and he's just not done it.
"He's got a great opportunity tomorrow to put things right and put in a big performance. I would love to see him start alongside [Luis] Suarez tomorrow but obviously Kenny [Dalglish] sees the team all week and he knows best."
Liverpool are hoping to reach their first FA Cup final since 2006 when they take on their local rivals at Wembley Stadium.