This is a simple feature that'd be really helpful. As far as I know, you can't allow users to do this without modifying XenForo files. I'd appreciate if it'd be possible for users to edit their own birthday if they have x permission, there'd be an option to let them modify it only once or infinitely.
Why do you want to let them change their birthday? Sure, it's possible that a user specifies the wrong date when signing up, but it's not something that happens every other day... If they want to have their birthday changed they'll contact you. You can change it through the ACP.
I help run a forum (http://www.redwarfare.com/) in which a lot of kids/teens use, many of them feel inclined to lie about their age to avoid getting blocked out by an age gate (I.E. you must be over 18 to register).
This isn't a huge issue but we do receive requests quite often from our users for their birthday to be changed.
Then it sounds like you have an issue with how people are registering, if people are asking for an age lower than the 18 years you have setup look into changing that. Would get rid of the problem of having to change a lot of birth dates.
For sites where age is significant in the registration process this will always be a factor. Since for most sites, age isn't an issue then I'd say this falls outside the scope of normal functionality and you'd be best off asking waindigo or someone else to code up a mod for you. To be honest, this wont even be a complicated mod.