So, the Monk's Martial Arts say:
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
Meanwhile, the Tavern Brawler feat says:
Your unarmed strike uses a d4 for damage.
My question is, could either a Monk or a Tavern Brawler choose to deal the flat 1 damage an untrained unarmed strike would do, instead of the d4 or better die?
Best Answer
Monk: Yes.
Monks can choose to use their attack's original damage or to use their Martial Arts damage dice. This often comes up since at lower levels, many monk weapons do higher damage than the Martial Arts dice can. As your quoted section says, monks can roll the Martial Arts damage dice instead of the normal damage.
Tavern Brawler: No.
Tavern brawler outright changes your damage dice for unarmed strike to 1d4. It doesn't have the language that Martial Arts does of allowing you to choose which damage amount to use.