I think the author has chosen this name, because he wants you to think Dr Jekyll and Mr Hyde are two different persons. First you read about Dr Jekyll and later about Mr Hyde, it looks like they are different persons, but they aren't. Actually, Dr Jekyll is the same person as Mr Hyde. Dr Jekyll has used a kind of drug, which changed him into another person ( Mr Hyde ).
The story is about Dr Jekyll, Mr Hyde, and Mr Utterson. Mr Utterson thinks Hyde is blackmailing Jekyll. When Hyde beats up a girl, he uses Jekyll's money to pay the family. Utterson tries to solve this mystery. Utterson goes to a friend of Jekyll, this is Dr Lanyon, but he says he hasn't seen Jekyll for a couple of years because couldn't agree on medical thing. On his way to Jekyll, he meets Hyde and asks him where his house is. Mr Hyde looks very angry and tells him his address in Soho. Then Utterson goes to Jekyll's house and talks to Poole, he is the butler who says that Hyde has a key to the house and is free to come in when he wants to. After a year girl sees Hyde kill Sir Danvers Carew in the street under her window. The next day the police comes to the house of Utterson to ask him to answer some questions about the body, because the dead man had a letter for him in his pocket. When Utterson hears that Hyde killed the man, Utterson takes the police to Hyde's home. Jekyll has been acting strange weird since Hyde begins to come to his house. After a while Jekyll tells Utterson that Hyde has gone forever and they will never see Hyde again. Utterson goes to Lanyon to ask if it is about Hyde again, but Lanyon is sick and dies. He left a letter for Utterson, it has to be opened when Jekyll dies or disappears. After a few weeks comes Poole to the house of Utterson and starts to ask him to come with him to the house of Jekyll because he thinks something strange is going on. Poole tell Utterson that Jekyll sends him to a lot of different chemist shops buy medicine but always he sends it back because it is not the good medicine. Poole thinks that Hyde is in the laboratory and that he murdered Jekyll, so he wants Utterson to come and breach the door to see what he is doing. There they find Hyde dead on the floor of the laboratory and another letter for Utterson which explains the Jekyll's strange behavior. The letter tells Utterson to read Lanyon's letter. The men also find a new will written by Jekyll that says he gives everything to Utterson. It seems that Jekyll has discovered some potion that can change him into Hyde, the evil side of Jekyll. Jekyll gets addicted to the potion and uses it more and more often, until once he turns into Hyde without the potion. It becomes more difficult to change back into Jekyll because his bad side is taking over his body. He sends Poole to buy more of the potion, but he cannot find any more of the same medicine. One night Hyde goes to the house of Lanynon and drinks the potion in front of his friend. Lanyon sees Hyde change back into his friend Jekyll. He is shocked, because of that he becomes very ill and then dies. At the end when Utterson warns Hyde that he is going to breach the door, Hyde drinks the poison and kills himself because he knows his evil side must be stopped.