When to Allow Makeup Exams
One good way to approach this kind of thing is to state a policy in your syllabus. For instance, you could have:
Make-up exams will be given only in case of an emergency or unavoidable problems such as an illness requiring hospitalization. You must communicate with me as early as possible about the problem.
Under this policy, you would just tell the student no in this situation. Or you could craft a policy that does allow a make-up exam in this situation.
The advantage of having it spelled out in your syllabus is that it's communicated clearly, it's clear why you're making the decision you're making, and administrators will support you in your decision.
Personally, I just say no in the kind of situation you're describing. The student is making a commitment by taking the course, and the nature of the commitment was made clear from the start in the syllabus. It's quite a bit of extra work for me to write a different exam for one student and make arrangements for it to be proctored. When you're a college student, it's simply not reasonable to expect to have a week off from school with no consequences. Lots of other students would like to have a week off with no consequences.
You have to consider carefully whether to allow this. If you do, certainly I would make it be before the class. But of course it is completely your decision.
As far as setting a precedence, there is no way the precedence will affect other faculty. If a student comes up to me, for example, and says "but Prof. X made this allowance for me in the past", my answer is simply "I don't care". But you have to worry about the precedence you set for yourself. What if another student comes to you at the next exam and wants to reschedule to take a vacation? Or if you have an exam right before Spring Break, 3 or 4 students want to leave town early and reschedule their test?
In short, if you make this allowance for one student, you are committing yourself to make this allowance to any student who asks, for at least the rest of the semester. I wouldn't want to put myself in that position.
It sounds like you want to do this, but have a few concerns: fairness to other students, inconveniencing yourself, and setting a bad precedent.
If you have the student take the exam early, then they don't get any sort of advantage over the other students. No one is going to be jealous that a fellow student took the exam early. There is a chase that the student will blab about the exam to other students. However, I don't think you should punish everyone because a few people are dishonest. Tell them explicitly that you expect them to keep the test confidential until after everyone else takes it, and you should be fine there.
Most universities have a proctoring system set up specifically because it's a common thing for students to take exams outside of the normal time, and using that system shouldn't be a real inconvenience. If you feel you can trust the student to keep the test confidential, then you don't need to create a separate exam. If they're taking it early, then you can grade it with the other exams. I don't see why this should be a burden on you at all.
Most students are not going to be willing to take the exam early, so I don't see this becoming an epidemic that you need to worry about. If other instructors want to be sticklers about following the schedule in their syllabus, I doubt your "precedent" will phase them any.
In short: there's no real reason to tell them no. It's not going to cause any problems. If they asked respectfully, then let them take the exam early.
FYI, I'm answering this from the perspective of a former student that has taken several exams early for similarly "trivial" reasons. None of my instructors that I ever asked have ever batted an eye at it. There was one time a fellow student of mine wasn't allowed to take an exam early, and the reason the instructor gave was "it'd be unfair for him to take it early." We all thought it was BS and unreasonable to turn down his request. So, in my experience, the most common and accepted practice is to allow these sorts of requests.