From ToEE

A luck roll is when a player can choose to 'get lucky', and ask for a dice-roll to be made again. These can be real lifesavers if used at the right time. The following rules apply to luck rolls:

• Luck rolls are a special reward to players for having made significant contributions to the game. All of the following will be considered:
  • Additions to storyline and flavour (eg, character blogs and histories).
  • Maintenance and organisation of characters, infomration, or schedules.
  • Artwork, diagrams, and drawings.
  • Anything else that contributes significantly to the atmosphere or enjoyment of the game.
• Luck rolls are awarded by the DM at the start of a session. Luck rolls are 'single-shot' rewards, and must be used during the session they are awarded, or lost.
• Complaining about how you haven't received a luck roll is a great way to continue not to receive one.
• A player can only invoke a luck roll for an event that is targeted at their character. These events can include saving throws, to-hit and damage (both when attacking and defending), healing rolls (both when receiving and casting), stats checks, and so on.
• Luck rolls cannot be used for events that affect the party in general. You can't use a luck roll to avoid a wandering monster, or to re-roll party initiative.
• If a luck-roll is made, then it's as if the first dice-roll never happened. This means that if your second roll comes out worse, then tough, you have to live (or die) with it.
• Certain dice-rolls are ineligible for luck-roll usage, including (but not limited to) gambling with Furnok at the inn, and dicing for treasure shares with other characters.