Arsenic

In reality it was a mixture of different drugs, arsenic was just one of them, concocted to keep Edward from dieing. Whatever they gave him, apparently, kept him alive for some time.
It's hard to say if the arsenic played a part in keeping him alive, but it was likely something else in the mixture. Arsenic, even if it does help treat certain ailments, is more helpful in the long run to a persons internal organs

It was actually commonly used in a lot of drugs, though is now mainly used to treat certain types of cancer

you have to remember that they were shooting in the dark in those days. They decided on what was a medicine by experimentation, and if you took it and did not die immediately then it must be good for something. Arsenic was and is still being used in small does to treat illnesses, but in those days it was pure coincidence that it treated the disease. Basically small doses will kill off small things first such as bacteria. We do something similar with anti biotics. If we took them full dose it would kill us, but a small dose for a short length of time seems to help the body fight the bacteria. The big problem comes when they throw it at things that are not bacteria bases. Then it equals poison to all of us.

Arsenic was used to treat syphilis. Wasn't it believed at one point that Edward died of syphilis passed down from his father? Perhaps the screenwriter labored under that impression.