In the end you, your dad, and your brother should come to an agreement so there's no surprises or unfulfilled expectations, but here's my opinion:
If you can afford to make the additions now:
I would offer to pay fully for the addition, with the understanding that the additional value that it generates is yours. That keeps everything in your name, and should be fair since you pay for the expense and someday reap the benefit.
If you can't afford to make the additions now:
I see two options: have your brother buy your father's house, giving you half of the proceeds, and use those proceeds to make the addition as above, or split the cost of the addition and have some sort of contract drawn up promising to reimburse him (with the amount of the reimbursement very clear, like XXX dollars plus accrued interest at Y% annually) as a condition to selling the house.
One other part you didn't mention is any compensation you get for keeping your father at your house. What compensation (if any) you get is not as important as making sure that the three of you all agree on what is fair.
In any case, clear, honest communication and full agreement is key. There is a very real risk that when your father's estate is settled that there will be disputes over what the agreement was and who it entitled to what. Having everything in writing may sound cold, but it keeps everyone on the same page.