first imagine an upturned box that is big enough for your beehive to sit on. (say about 8 inches tall).
Now sitting on top of this box is a platform that overhangs the box by quite a reasonable extent (maybe a couple of feet).
This platform is then attached to the edge of the box by means of hinges. So if you were to stand on the platform and walk past that pivot point, it would tip.
Underneath the platform (on the far side to the hinges) is a microswitch that will detect when the platform is tipped.
You can now place your beehive on the platform over the box.
In the meantime, underneath that overhang is a very heavy weight. This heavy weight is suspended beneath the overhang on a runner. It can also be moved towards or away from the pivot point by means of a lead screw.
So all the arduino needs to do when it takes a measurement is draw the weight all the way in unitl it finds the end stop. Then slowly move the weight (by turning the lead screw) until it detects that the microswitch has been tripped.
From calibrations carried out before you deploy this gizmo you can then use a lookup table to see what weight that distance implys.
Naturally, you could have a saftey catch to prevent the platform rising any more than a couple of millimetres, so there's no danger of the beehive ever being tossed off.
If you like I could probably knock up a diagram in paint to explain better.