First, the click is normal -- it's the shift interlock that's part of the cars of that ilk (the last years of the 240) -- and is nothing to do with the problem.

The problem is, based on your report that there has been no loss of fluid whatsoever from the reservoir, most definitely the master cylinder. In fact, I can tell you exactly what's wrong.
There is a seal on the internal piston that is supposed to confine a volume of brake fluid from the reservoir as the pedal (and the piston) first moves -- that confined fluid is then pressurized and that pressure is transmitted to the calipers. As the pressure builds, it resists the pedal's effort to a point when you cannot depress the pedal any farther.
But if that seal fails (as it has in your daughter's master cylinder), brake fluid will leak past it back into the reservoir -- in a sense, the fluid is simply recirculating back to the reservoir -- and as it does, pressure cannot build up that will resist further pressing of the pedal. Thus, you can now press the pedal all the way to the floor.

Replace the master cylinder.