Is the belief of a person's family in "there is no god but Allah and that Muhammad (peace be upon him) is His Messenger" sufficient for him to enter Paradise?

Uttering the two testimonies (Shahadatayn) is not sufficient for entering Islam. Rather, their conditions must be fulfilled. The pillars of Islam are belief, utterance, and action.

The two testimonies are not valid unless they are based on knowledge, certainty, sincerity, truthfulness, and action according to their implications.

The conditions for "La ilaha illa Allah" (There is no god but Allah) are seven: knowledge, certainty, acceptance, submission, sincerity, truthfulness, and love. These conditions also apply to "Muhammad Rasul Allah" (Muhammad is the Messenger of Allah).