The Australian led Spurs to the Europa League title this season - a first UEFA trophy in 45 years won by the club. The triumph also sees Tottenham handed a place in next season's Champions League.
But Tottenham's 17th place in the Premier League has cost Postecoglou his job, despite the squad's injury crisis throughout the campaign.
Son posted to social media: "Gaffer. You've changed the trajectory of this club.
"You believed in yourself, and us, since day one and never wavered for a second. Even when others did.
"You knew what we were capable of all along. You did it your way. And your way brought this club the best night it's had in decades. We will have those memories for life.
"You trusted me with the captaincy. One of the highest honours of my career. It's been an incredible privilege to learn from your leadership up close. I am a better player and a better person because of you.
"Ange Postecoglou, you are a Tottenham Hotspur legend forever. Thank you, mate."