I have a mtgox verified account, but I'm not sure if I can add a family members bank account to my mtgox account.
That's correct -- you cannot withdraw from Mt. Gox to someone else's account.
The other person is free to create a Mt. Gox account and then withdraw the funds that way, however there's no guarantee they won't be subjected to the requirement of having to verify an account.
There are other SEPA exchanges with less restrictions. I believe with BITSTAMP you can specify the account for a SEPA withdrawal at the time of the withdrawal, therefore you could enter a family member's account at that time -- presuming it is below their AML restrictions.
Another method is to arrange a local transaction. For instance, with LocalBitcoins.com you would arrange to have the family member pick up cash from a buyer. How that would work as a three-way transaction is where you would arrange everything, then release the bitcoins when your family member confirms that the buyer is there with cash in a face-to-face trade and then you release the coins at that time:
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http://www.LocalBitcoins.com