Henderson took plenty of criticism after taking over the armband from Gerrard in 2015.
But the England international lead the Reds to their sixth Champions League crown last season, and looks set to become the first Reds captain in 30 years to lift the Premier League trophy.
"When he first came in, it raised eyebrows in some ways, it was 'where is he going to play', we already had Steven Gerrard at the club, we had bought Charlie Adam as well," Carragher told Sky Sports.
"There were a lot of midfield players as well. It was Damien Comolli who brought him in, young England player, loads of energy. He found it tough early on because he was played out of position in right-midfield.
"The thing I love about Jordan Henderson and it's a little similar to my journey because at times I found it tough and people said I wasn't good enough and they have said that to Jordan Henderson. That just shows the character of who he is.
"You can use words like great professional, great character, and these are just words. What does it mean? It means the way he trains every day, people questioned why he was captain. I used to argue with Jordan all the time and I know the character and personality I want at my club and that is him.
"Who else could be captain after Steven Gerrard? I'm delighted and people like Jordan have something special about them, where you never write them off. At different times of his Liverpool career, someone has come in and you thought he might not play.
"[Alex] Oxlade-Chamberlain, central midfield, Naby Keita, big-money signing, young, lots of qualities but Jordan Henderson still plays. Playing for nine years at a club like Liverpool is testament to his abilities and personality.
"Whether player of the year or footballer of the year gets done this year, I'm not sure. I thought Sadio Mane was Liverpool's best player when the season stopped and I wouldn't change that view.
"In terms of who has been the outstanding footballer of the year, it's about more than what you do on the pitch, I don't think there's a more outstanding character than Jordan Henderson."