Well, the problem is that they insist on neither: that both the price stay the same and that the extra stuff is required. Rock - Me - Hard place.
BUT I do feel bad in these cases because I believe they thought it would be included. Plus it may truly be necessary for the final product to be useful for them. So I want to help them, plus, ya know, don't want them pissed at me :)
Personally, I'd say you just apologise for misunderstanding their intentions, politely inform them that the new feature is going to cost extra, and then suggest that, due to the misunderstanding, you're going to reduce your price for that implementation to $X, where $X is either the price you actually have in mind (if you don't mind being less than 100% truthful for the sake of fairness) or a slightly lower price (if you're willing to pay for their mistake so as to maintain more truthfulness). Yes, it can be hard to admit to mistakes you didn't actually make, but it's often necessary to not piss off some people.
BUT I do feel bad in these cases because I believe they thought it would be included. Plus it may truly be necessary for the final product to be useful for them. So I want to help them, plus, ya know, don't want them pissed at me :)