How should payments to children be handled?

In order to live near my son’s family we sold our house and built a house on their property next to their house. I handled the construction and paid for nearly all of the costs. The cost of the house was a gift to my son, so he owns the house. They have also requested that we give them money on an ongoing basis equal to about two thirds of what the house would rent for. They do not want this money to be considered as rent, but as ‘cost sharing’ for living there. Is it legally proper for them not to treat the money we give them as rent and not declare it as rental income for tax purposes?