Hamilton is leaving McLaren after six seasons with the team, the last three of which have been alongside Button.
The former has scored 10 wins in the period and has finished ahead of Button in the standings twice. Button meanwhile has won eight times since joining McLaren.
The former world champion, who finished the 2012 season in fifth position, says working alongside Hamilton has taught him a lot of things, and he feels his team-mate has also learned from him.
"Lewis is extremely fast and he definitely has that speed that not a lot of people have - outright pace over one-lap - but as we know, Formula One is not about a single lap. It's a race and that's where you score points.
"Of course, he's been a tough team-mate in terms of speed, but also a good team-mate to work with and really develop the car. I think he's learned a lot this year in terms of understanding that it's not just about one lap in terms of looking after the tyres.
"So I think the partnership has been good in terms of us learning off each other.
"Even if he says he hasn't, we've both definitely learnt from each other and it's helped us to improve as drivers."
In 2013 Button will be joined by Mexican Sergio Perez, who has competed in Formula 1 for two seasons, both with Sauber.
The Briton believes the role of team leader suits him perfectly.
"Definitely," said Button when asked if he was ready to lead McLaren. "It's not the first time I've done that. When I got to BAR and Jacques [Villeneuve] left it was exactly the same situation.
"I was there to lead the team and it's something I really look forward to and it's something where the best comes out of me in that situation. I can really build that team around me and direct the team in a direction I like with the car."
Button feels he is one of the best drivers when it comes to developing a car he feels comfortable with.
"We all drive differently and have different styles," he said. "For me I need a car I can develop beneath me and feel comfortable in. If the car feels neutral and unbalanced it doesn't work for me.
"I need to develop a car and engineer a car in a position that feels comfortable for me, and I don't think anyone can do a better job than I can in that position.
"The problem for me is if I can't get the car there I do struggle more than some."