Son, Spurs' top scorer this campaign with 15 goals and eight assists, took the captain's armband after Tottenham's record goalscorer Harry Kane moved to Bayern Munich last year.
Spurs also lost France's World Cup-winning goalkeeper Hugo Lloris, who joined Major League Soccer club Los Angeles FC after an 11-year stay in north London.
"It wasn't a big decision (on making Son the skipper)," Postecoglou told reporters before Tuesday's trip to West Ham United.
"With Hugo being a huge influence and Harry as well, Eric Dier too - there are others who have been at the club a long time. Part of it all along was that I felt like we needed to make a change to show we are going in a new decision.
"I felt Son was a natural and he has been a leader within his national team. I had no hesitation putting him in that position. Sometimes when you get the official title it can be a burden, but his performances have been outstanding."
The Australian coach's attacking philosophy has won much praise since he took the Spurs job in June and his team led the standings after going unbeaten in their first 10 matches.
Spurs are now fifth and three points behind fourth-placed Aston Villa, who have played a game more.
"We want to finish the season strong and keep building on the foundation of being a team that hopefully in 12 months' time people are looking at us differently and talking about us differently," Postecoglou said.
"To do that it can't just be about where you finish and how many points you get, it's got to be more than that. Obviously, if we finish strong, then we'll finish in a decent position, and we'll see where that takes us."
Postecoglou said he would keep working hard with his young squad.
"(Dejan) Kulusevski or Brennan (Johnson), Micky van de Ven or Destiny Udogie, or Pape Matar Sarr. All these guys are in their early 20s, in their first or second year in the Premier League. Our job is to improve all of them," he added.
