I would say steer clear. It's similar to accepting gifts and/or giving gifts. Eventhough it's may be innocent, it might be wise to use caution. To the previous responders, I don't think it's weird of a patient to request to be added on FB. These day many people are obsessed with looking on FB on the iphones, computers etc. Patients and families maybe totally innoncent when they want to be friends with you on FB. It is very important to set boundaries. Maybe you could set up a different facebook account for your "fans" (patients and their families, nurses, doctors, etc). Many famous people do it that way to help keep their private lives, private.
If you have family members on your FB page, you might want to used caution.
Apart from all that, when you take very good care of a patient at a point in their live when they are vulnerable, they may look at you as a very special person in thier lives and may want to be friends with you just because of that.
It happens to me all the time. So I had to set up a separate FB account for my "fans" where I can add patients, families, etc. I hope this helps give you some ideas.
I don't really like having to carry my picture ID on my uniform
with the whole world looking at my name and pic. Some looking me up on Facebook, right away!