Very unfortunate scenario, indeed, but I think the squeaky-clean thing is to give them a perfect score.
That is, if you screwed up, ... well, ...
EDIT: in light of other remarks... this presumes you do have a system to certify that the exam was handed in, so that students are not gaming you. Even in a large class, it's worth doing this. Having a list of names to check off can succeed even with a large class and in a large room, since you can take the viewpoint that there was no way for any exam to get lost _through_your_malfeasance_. Thus, in that situation, any missing exam is concluded to have not been handed in.
Then, subsequently, if someone's exam "is missing" (that was certified present earlier), you must have lost it.
Even if the student was not stellar, it would be presumptious (or worse) to exclude the possibility that they did well on the exam... Giving them the mean or any other quasi-uniform replacement score seems to me completely insufficient.
The fact that other students might resent ... your not having lost their paper? ... is a very secondary issue. I would think that, instead, they should understand that you feel responsible to the student whose paper you lost!