It is established in TNG Reunion that the arbiter of succession must choose the two strongest claimants to the chancellor's seat; after that, according to that episode, the two must fight for power.
PICARD: This is not a case of simple mediation. You are asking me to choose the next leader of the Klingon Empire.
K'MPEC: No. By tradition, the two strongest challengers fight for the right of succession. As mediator, only you can designate those challengers.
In TNG Redemption when Gowron is challenged by Duras's illegitimate son Picard must decide whether the challenge is valid.
TORAL: (as Picard is about to put the cloak on Gowron's shoulders) There is one. I will challenge him.
GOWRON: The illegitimate son of Duras cannot rule the High Council.
K'TAL: The Arbiter will consider his validity. Len'mat.
Picard ultimately decides that the challenge is not valid. Subsequently civil war breaks out between the Duras and Gowron factions. My question is therefore: why? Why not accept the challenge as valid and let Duras's son be easily defeated by Gowron (an adult and an experienced warrior)?