I can think of one thing that will be helpful to do. You can make an offering to the Monastic Sangha with the resolve to purify your mind and make much merit, and dedicate the merit to your relative. When you next speak with or meet your relative, you can tell him that you donated money to charity in his honor, and that he should rejoice in the goodness. Many Buddhist monasteries and organizations are registered charities worldwide.
It is a true statement that may not offend him. If you do not wish to disclose the identity of the Buddhist monastery, that is fine. If you wish, you can donate to other organizations as well, so that if asked "Where?" you may respond, "To X, Y, and several places," without mentioning any Buddhist organizations.
I did something along those lines when a relative of mine, a devout Christian, was very ill as well. It seems to have been of great help.
In addition, you can provide comforting words, reminding your relative of the goodness that he has done for you and others throughout his life, of the help that he has provided, and of wise and wholesome thoughts. You do not necessarily have to mention Buddhism. I think that a lot of Dhamma can be presented as secular. If asked, you can say something like, "I learned this from a very wise person."
These things will likely positively affect your relative's mental state in this life, as well as his future rebirth. I wish you and your relative the best.