This is said more often than I'd like, followed by, “Could you come up with something less expensive?"
My first thought is usually, "Sure could." Warning! This is a trap! Think about it. What attracted this person to your work in the first place? How much of that are you going to have to push off the table in order to fit their budget? How happy are they going to be with the final result?
Now and then, you might come up with something so brilliantly creative that the cuts you had to make in time and materials cost are offset by the design itself and, if you are really lucky, the client realizes this and goes away happy. But more often than not, they can clearly see that most of what attracted them to your work in the first place is missing from their project and, even though this was understood at the outset, they will not be satisfied with the result.
Faced with this situation, it might be best to simply realize that you are going to be the one holding the short end of the stick and accept that. I have done this on occasion. There have been times when I just needed a job in the shop to make payroll or whatever and I was willing to do a job that was cut to the bone. But thinking that you will have a happy customer when they are getting none of what brought them to you in the first pace is just going to give everyone a big headache.