Choosing good passphrase / password is quite tricky and it depends on your attack scenario (such as how much money/time attacker willing to spend and how well do the attacker know about you).
Your passphrase choice is vulnerable against dictionary attack, but if you think the attacker won't have idea about you and your family, then i think it's strong enough against weak/low cost attack scenario.
Would 6 random names be safe enough as passhprase against brute force attacks if my private key is known?
Your passphrase doesn't matter if private key of your Bitcoin address is known