I am a fan of in-game solutions when it comes to deck building games. Doomtown had a cool mechanic called Grifter, where players could start a single card with that keyword, whose ability would trigger at the beginning of the game only. Some grifters had mulligan abilities (both full-hand flush and select cards), others had thematic ones, while still others had economic incentives. These cards could all be costed accordingly and forced creative deck-building decisions. The most utilitarian Grifters were allowed across factions, while the more thematic ones were faction specific.
Not sure how this would translate into L5R (because we know nothing, as Crawd pointed out), but I could see a variation on the bamboo harvester theme allowing for neat design space: choose a holding which grants either mulligan, economy, honor, special ability, etc. The point being that mulligan could become a cost/benefit analysis as opposed to a structural guarantee.