Building a small web app just for fun is a great team building idea.
Helping your team break out of their normal work environment and challenging them to learn new skills will improve morale, empower your team and result in unexpected benefits.
Questions Answered:
Are team building exercises always necessary?
How do you get your team onboard?
How much planning should take place beforehand?
What happens to your real work?
What goals and timescales should be set?
Is it important to challenge your team with "uncomfortable" tasks?