But this is then an extra place where his credit card details are stored, and he may not have decided to trust Stripe like that yet.
The checkbox is whether you want to be able to pay again without retyping it.
Now having read this thread I get that in fact Stripe is storing it, but if I hadn't read the responses to my initial post, I'd just assume Kickstarter was asking to store my CC# which I wouldn't agree to.
The token can only be used with the private / public key pairs that Stripe provided to them, so even if a hacker got access to Kickstarter's database, they would still need the private/public keys to make use of the tokens.
Also, my expectations are that only whitelisted IPs should be able to access Stripe with the key pairs of Kickstarter.
Basically, the only important information Kickstarter gets from any card are the last 4 digits, whether it's Visa, Master Card, AMEX, etc., and the expiration date.