ThinkFun, the US publishers of Rush Hour, wrote a blog post about this when they made the mobile app with a bunch of autogenerated levels maybe around 2010. Unfortunately that article is basically impossible to find now, since they redid their website and deleted the old posts. (I say "impossible", since I spent a couple of hours looking for it again while reading papers on procedural puzzle generation last year. Maybe somebody here has better luck).

The basic metric their level generator used for quantifying interesting difficulty was the earliest point of non-trivial divergence in solutions. I.e. if there's a puzzle with a 50 and 51 move solution, having those solutions diverge on move 3 is interesting. Having them diverge on move 45 isn't.