Stones, 20, committed his long-term future to the club by penning a five-year deal - a reward for his stellar 2013-14 Premier League campaign.
And the former Barnsley youth player, who made 26 appearances in his first full season at Goodison Park, told Everton's official website he has the potential to wear the armband just like skipper Phil Jagielka.
"I'd love to captain Everton," said Stones, who has also earned two international caps for England.
"It's a massive dream of mine and the big name players that have done it in the past, I would love to follow in their footsteps.
"I'd like to think that I have the attributes [to be a captain].
"I'd like to think that I can do it in the future and, if that chance came along, I would jump on it straight away."
Stones added: "It's been massive for my learning curve to play alongside Jags [Jagielka] and Sylvain [Distin].
"They are experienced pros and to learn off them day in and day out and in games is helping me as a person.
"They are leaders and hopefully I can keep learning off them to get better as a player."
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