The plots will be the same, but just make up the names.
Will this effect my essay score?
It won't affect your score. The reader will give you credit for what you do well. I would caution you against using a book that you don't know well enough to remember the character's name. Come to the test prepared, having read several different books and knowing them well. That way, you will do well on the essay no matter what the question is.
I think your best bet is to refer to the character by descriptions if you forget the name. Call him "the protagonist" or "the lawyer." Show off what you DO remember about the book rather than what you do not.
This is a tough call. I think that it is better to concede that the name of the character escapes you. It seems really silly that you would try to make up a name for a work of literature. I cannot imagine that this would help your score. I think it's better to concede that the exact name of the character escapes you "at this time," and go on with your essay. It certainly seems like you are writing about plot or writing about other elements that the name of the specific character is not the only thing that you are writing about, so I think that being able to simply admit that you forgot the name cannot be a bad thing. Remember, you are "showing, not telling," which might allow you a bit of leeway in the exact names of characters. I think that it is better to accept your responsibility on this one and not try to pretend there is no problem by coming up with an odd name. If the rubric is designed so that the overall paper is assessed, perhaps, the deduction might not even be that noticeable for it.
You can probably get a way with it. After all, tests like the SAT don't dock points for inaccurate facts and figures. I think its better to write along the lines of "The main character"...etc because that way we know who you are talking about and it doesn't make you sound silly.