Why wasn't Solar Impulse 2 cleared for takeoff?
Interesting question. The ATC orders mention only two reasons that controllers have to say "at your own risk":


The pilot "insists" on using a closed or unsafe runway (3-3-2)
A helicopter takeoff or landing uses a non-movement area (3-11-2, 3-11-6)


Obviously Solar Impulse 2 isn't a helicopter, so that leaves the closed or unsafe runway reason. I don't know for sure why the tower used the "own risk" phraseology - it would be great to hear from an actual controller - but some possibilities are:


The runway was marked as closed (for Solar Impulse 2) on the tower controller's displays, and he said "at own risk" more or less out of habit
There were vehicles on the runway, therefore it was "unsafe"
Solar Impulse 2 is such an unusual aircraft that the controller felt he couldn't contribute anything to the safety of the takeoff, so he was simply saying "go for it, I can't do much to help you"


Of course, the problem with all those suggestions is that they should apply at all the airports, not just half of them. But I did find this quote from a controller on the AOPA forums (members only):


  Although it's normal for helicopters, I issued this "departure will be
  at your own risk" to an aircraft once. The runway was open, he wasn't
  departing from a taxiway or anything like that. The problem was fog,
  where I couldn't see the runway from the tower, and therefore I
  couldn't determine whether the runway was actually "clear" for his
  takeoff.
  
  I don't know if this is the right response in this situation or not,
  but I was a pretty new controller at the time, hadn't faced this
  situation before, and figured the "at your own risk" was appropriate.


That could have applied to the Solar Impulse situation: with other vehicles and personnel on the runway, the controller can't say it's "clear". And just like the controller on the forum, perhaps some of the controllers handling Solar Impulse 2 felt that the non-standard wording was appropriate for a very unusual situation. The other controllers who did issue a takeoff clearance presumably felt differently.