Just Tell Me When We’re Dead! by Eth Clifford, 1983.
This is another book in the Mary Rose and Jo-Beth series.
Mary Rose and Jo-Beth are visiting their Grandma Post with their father while their grandmother is getting ready to go into the hospital for an operation. Their cousin Jeffrey, an orphan who lives with their grandmother, is supposed to come and stay with them until their grandmother is better, but he doesn’t want to go.
Jeff has never really gotten over the hurt from when his parents died. They used to travel a lot, but they would never take him along because he was too young. Then, one day, they were killed on one of their trips, which is why Jeff now lives with his grandmother. Now that his grandmother is going to the hospital, Jeff is afraid that she will die, too, and unable to face that, he runs away to be on his own.
The first place he goes is to an island in the middle of the lake near his house. The island has campgrounds and an amusement park, which is now closed for the season. Mary Rose and Jo-Beth, realizing where Jeff has gone, follow him there. But, the children are not alone on the island. When Jeff is captured by two criminals who are looking for loot that they stashed on the island years before, he has to keep his wits about him to find a way to summon help. Meanwhile, Mary Rose and Jo-Beth have no idea what they’ve just walked into.
At the end of the adventure, Mary Rose, Jo-Beth, and their father help Jeff to make peace with the loss of his parents and to understand that, even though unexpected and scary things can happen in life, his parents never meant to leave him. They loved him, and his grandmother and other relatives also share his sense of loss.