Just like how a user can prove they are the owner of a Bitcoin public key by signing a message, how can a Monero user prove they are the owner of a Monero public key?
Best is to sign a message consisting of your address, with your address. The sign and verify commands have been merged just yesterday: https://github.com/monero-project/bitmonero/pull/818
First, make a file with the data you want. It just happens that, when creating a wallet, your address is saved in WALLETNAME.address.txt.
Then send the signature output by that command to the other party, who can now verify:
verify WALLETNAME.address.txt 4xxxxxx SigV1xxxxxx