Is it permissible for one observing I'tikaf to contact his family by phone to check on them a few times, and not within the mosque's prayer area?

There is no harm for the mu'takif (one performing i'tikaf - spiritual retreat in a mosque) in contacting his family to check on them, even if his call is made from inside the mosque. This is indicated by the fact that the Prophet, peace and blessings be upon him, used to perform i'tikaf, and his wife Safiyyah, may Allah be pleased with her, would visit him at night and speak with him. Ibn Qudamah said: "There is no harm in speaking for his needs – meaning the needs of the mu'takif – or in conversing with others." However, the mu'takif should busy himself with acts of worship and avoid that which does not concern him.