Depending on the extent of the redesign, I wouldn't be surprised to see the Five Rings themselves as the victory condition(s). Something like trying to control X Rings before the opponent, or place X tokens on Rings. Different clans would have different rings that they are good at. This should help balance both the "downhill" effect of military and the stalemate aspect of honor v dishonor.
Having one "turn" where both players move through the phases together should help address the going first advantage.
One rule change I think is almost certain is that Unique will act more like Singular, where you can have a full amount in your deck but only one in play.
I could see some fun ways they could adopt that. Each ring traditionally has a correlation to some aspect of the game, though it's changed a fair amount over the history.
I could see Fire being dueling.
Air being political.
Earth being defense.
Water being offense.
and Void being either magic or something to do with card usage/draw.
Of course there are other ways to slot them, but they could hit all the bases of the clan strengths and weaknesses.