I believe that the Darksaber should have actually rightfully passed to Paz Vizsla! In case you're unclear on exactly who that is, Paz is the heavy infantry in the first season of The Mandalorian and disagrees with Din Djarin on several different points. So this points out a

potential plotline for the third season: Din Djarin, who is in current possession of the Darksaber, is challenged for it by Paz.

So this altogether means that either Bo-Katan is aware of Paz Vizsla and his potential threat to the throne or it could be that Rebels is about to get retconned. And of course given Disney's willingness to do anything to force their canon onto the helpless fans, well, this could very well happen. It could also be that Bo-Katan does not want to fight Din (there does seem to be a bit of a romance blossoming in my opinion) or simply does not want the darksaber anymore. Of course a final theory states that Bo-Katan is trying to get Din to become more like her and the other Mandalorians, and attempting to have him leave behind his history as a Child of the Watch. Of course

this also could have been remedied by Din taking off his helmet to say farewell to Grogu at the end of Season 2.

However, it's feasible that Bo-Katan sees Din as a worthy leader and doesn't want that to go to waste. My favorite theory, though, is that she's using it as an excuse to keep him around so that they can retake Mandalore.

I hope that answered your question. Let me know which theory is your favorite or makes the most sense.

(P.S. Knowing Disney, the issue of Din possessing the Darksaber will be resolved in the first five minutes of episode 3x1.)

I believe that the Darksaber should have actually rightfully passed to Paz Vizsla! In case you're unclear on exactly who that is, Paz is the heavy infantry in the first season of The Mandalorian and disagrees with Din Djarin on several different points. So this points out a

potential plotline for the third season: Din Djarin, who is in current possession of the Darksaber, is challenged for it by Paz.

So this altogether means that either Bo-Katan is aware of Paz Vizsla and his potential threat to the throne or it could be that Rebels is about to get retconned. 

It could also be that Bo-Katan does not want to fight Din (there does seem to be a bit of a romance blossoming in my opinion) or simply does not want the Darksaber anymore.

A final theory states that Bo-Katan is trying to get Din to become more like her and the other Mandalorians, and attempting to have him leave behind his history as a Child of the Watch.

Of course this also could have been remedied by Din taking off his helmet to say farewell to Grogu at the end of Season 2.

However, it's feasible that Bo-Katan sees Din as a worthy leader and doesn't want that to go to waste. My favorite theory, though, is that she's using it as an excuse to keep him around so that they can retake Mandalore.

(P.S. Knowing Disney, the issue of Din possessing the Darksaber will be resolved in the first five minutes of episode 3x1.)