Wilson skippered Hearts to the Scottish Championship title last season before joining Gers on a three-year deal this week.
Rangers boss Mark Warburton has been talking to former Watford and Derby skipper John Eustace with a view to bringing the 35-year-old to Glasgow as a mentor for a more youthful squad.
But Weir believes 23-year-old Wilson has the attributes to follow in his own footsteps and wear the armband.
The 45-year-old, who played alongside Wilson when the centre-back first broke into the Ibrox team as a teenager, told RangersTV: “I think you forget how young he was when he was playing here the first time and, obviously, he’s gone down south and had various clubs there.
“He played for Liverpool and he was out on loan at Bristol [City] and a couple of other places. He’s learned a lot and he’s matured a lot.
“Meeting him in the last few days again you can see he’s grown up and he’s developed. He’s a different lad to what he was, although he still has a lot of the same characteristics. He’s a nice person, which helps as well.
“As I said, he does tick a lot of the boxes and another one is the fact he is vocal, he is a leader and he’s a potential captain. He’s one of those types that are few and far between and that’s another added bonus with getting Danny here.
“He’s got the attributes to come in and be the right type of person, set the tone for everyone else and be a part of what we want to do here.”