Today put a little bit more thought into solution 1 (see last project log entry, the one with rack and pinion). I think I will go with a rack, which slides in a tube and rotates a small gear. Connected to this gear is the rotor of an axial flux generator. This type of generator are well suited for low RPMs, you can see the principle at the end of this project log post.

In the following you can see an overview of the design I came up with. Point A and B are connected to the trees and the linear power generation unit is in its fully compressed state:

A more detailed view into the core. Please ignore the design of the housing :-). On the left is the tube where the rack sits in during full compression, in front you can see the pinion which is connected to the (transparent) rotor with 16 permanent magnets. Behind that is a red stator which holds 8 coils. The size and number of magnets and coils is not defined yet, this is just for illustration.

Last but not least the difference between radial and axal flux in generators: