The Addicks are managed by former Walsall player and coach Dean Holden, who worked with Dobson during his time at the Bescot.
Holden told charltonafc.com: "“George is a player I worked with at Walsall a long time ago now.
“I know exactly what he’s about as a character and as a person. He’s infectious, he’s 100 per cent in every training session, every game and that can’t be doubted at all.
“There are a number of other players as well that could have taken the armband but I just felt that George was the right option.”
Charlton drew their first game with Holden in charge, 1-1 at home to Perterborough United on Boxing Day.
Dobson, aged 25, made 81 appearances for the Saddlers in two spells, a loan between 2016-17 and on a permanent contract between 2018-19 before leaving for Sunderland.
Meanwhile defender Peter Clarke has extended his loan spell at Oldham Athletic until the end of the season
The 40-year-old, initially joined the club in November, made his 900th career appearance against Torquay earlier this month, and has made over 100 for The Latics.