It depends on which client you're using, but on the web:
1. Open the board.
2. Click on the board name itself (in the upper left).
3. A rename popup should appear.
This only works if you're an admin on the board - that'll be true if you created the board, or someone else added you as an admin.
I have just started joined as an admin with one other and have added a board, but cannot change the name - no amount of clicking on the board name on any screen allows me to edit as most things are done in Trello.
Are there more rules to being allowed? Perhaps the original owner/admin is the only one who can do this?
Win7 Pro | Chrome
In addition to Dan's answer, if you are on an iOS device (probably similar for Android as well), if you:
1. Tap on the "..." symbol in the upper right of the board view
2. Tap on the gear icon in the upper right
3. Tap on the 'Name' field and edit the name of the board
that should rename the board on mobile.
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