What is the benefit of this action?
Benefit of online wallet or benefit of signing messages?
Oh sorry, the signing of messages
A signed message means the key owner approved the message. The frequent use is to sign payment authorization, and this is transparent inside the wallet.
Another use might be with a configuration message. If I change my payout at Eligius I can set the new donation amounts or thresholds. When I sign that message, the Eligius software knows it was really me, and not some random griefer on the internet.
Another message that might be signed would be "Send $500 USD to John Doe, 123 Main St, Dallas in payment for 0.75 BTC"
The main benefit is that you have to do it in order to set up Namecoin payouts in the Eligius settings. If you don't set it up, your Namecoin earnings are forfeited to the pool.