Surely we can say that an electron isn't the wave function but naturally there won't be the wave function without an electron hence bit is it and vice versa. Let's focus on the question you made.
First of all, we have an electron. Its mere existence is an information. It's an elementary particle. Based on how other elementary particles like photons or neutrinos are created (according to ToEbi) we can say that electrons are created within a high density of FTEPs. Because the great amount of electrons there has been a circumstances where this caliber creation volume can happen, The Big Bang it is.
Back to the box... because the high spinning rate of an electron it's highly unlikely that our electron is stationary in the box. It's just bouncing around (depending of course on wall material). Each contact with the walls reveals more information in form of photons. By tracking these trails we get a pretty good information on the velocity, energy, spinning rate, spinning orientation and path of the electron, within Heisenberg uncertainty principle.
Did my answer help at all?