It used to be only athletic teams that had logos on clothing. Now work teams, families, all kinds of groups have t-shirts, caps, bags, jackets . . . A logo could be long term for a team, or a special one could be designed for each big project (perhaps with a consistent shape or wording).
Show off team pride; help members identify as a unit; use the making of the logo as a team building activity
Determine who will make the final decision on the logo - e.g., a special committee or all members via vote. With a large group, a committee could narrow down the submissions to a small number (3, 5).
Ask all members to submit designs for a team logo. Graphics can be drawn on paper or created electronically. Specify any parameters: e.g., must fit on a t shirt; must display well in either black and white or color Display all submissions for all to enjoy and discuss. Print garments or objects with the logo.