I just found this on the wikipedia page for Farkle, as play variation:

In a variant described as "piggybacking" or "high-stakes", each player after the first can choose to begin their turn either with a fresh set of six dice, or by throwing the dice remaining after the previous player has completed their turn. For example, if a player banks three 1's for a score of 1000, the next player may choose to roll the remaining three dice. If they score at least one die, they score 1000 plus whatever additional score they accumulate. Players may thus assume the greater risk of farkling for the chance of scoring the points already accumulated by the player before them. If a player ends their turn on a "hot dice", the next player may "piggyback" using all six dice.

So maybe piggyback or high-stakes would be a good name for the rule?

I love this game! I’m so glad y’all made it for Playdate. I have a feature request if you’re taking those.

We have a family rule where if you choose to stay the next player has the choice of picking up where you left off or starting with their own set of six. For example say my running score is 1200 and I have two dice left to roll but I decide to stay. The next person can choose  to roll those two dice. If they get something valid (say a 1) their running score would be 1300 and they could choose to stay or roll.

Any chance you could add this rule in as an optional rule ?