Step 1 - Set the tone - Explain to the team that you will live the team brand with guidance from your core values and purpose statement.
Step 2 - Ask the question - Ask the team to share ideas, images, names, animals and other words that represent the core values and the purpose. Typically, brands will jump out at you, as the team pattern-match the words.
Step 3 - Capture the ideas - Take note all of the suggestions. You will naturally reach a point where people run out of ideas, so it's up to you to keep the conversation going with thoughtful questions.
Step 4 - Ask more questions - Because you already have core values agreed upon, ask the team "What's the first image that comes to mind when I say [core value]?" You'll end up with 15-20 names on the board.
Step 5 - Refine - Whittle down the names on the board by asking "Who's 100% on board with [this name]?" and cross off the names that don't fit. You'll hear groans and see indifference in body language, so use your group's feedback as a guide.
As the facilitator, it's your role to dig into contention and disagreement, to make sure everyone is all in on the final name.