Bring a photo copy of your passport. It’s all you’ll need as identification when you walk around Havana or any other large city. However, when you need to cash travelers’ cheques or get money from a credit card or rent a car or check into a hotel, you’ll need to show your actual passport, but at all other times, a photo copy of your passport is adequate identification. You also need to carry your passport when you go from one province to another.
If you stay in a home stay, you’ll need to show your passport and Cuban visa to your host, who is responsible for registering you, within one day of your arrival at your home stay with the local immigration authorities. Your Cuban host will need to take your visa (but not passport) when s/he registers you at immigration, but will normally return this at the end of the day.