What can be done with a bitcoin besides exchanging it for goods or services? Some people have described a bitcoin as a commodity, but every commodity I see on this Wikipedia page seem to have a non-financial purpose, which also gives it a financial value.

What can be done with a "Bitcoin" (if anything) that has a non-financial purpose?
A Bitcoin can be used to permanently trace the association between different public keys.  For example a user can send a transaction to a second user and publicly and permanently link the accounts.  This may have utility in Web Of Trust scenarios.
The low value bits of an individual bitcoin can cheaply store metadata.
Saving them as a collectors item
Sharing (p2p style) a small amount of data