The author of Ethan Frome used the wrong spelling of "aid" (for aide) in the first chapter.
We need to be careful when leaping to such conclusions, especially when we consider the ways in which the English language changes in terms of its usage over time and how words come into fashion and go out of fashion, and the different meanings that a word actually has. It is highly unlikely that a classic work such as this one would be allowed to contain a spelling mistake such as you have highlighted. I suppose you are refering to the following example from the first chapter:
...when she entered the Fromes' household to act as her cousin Zeena's aid, it was thought best, as she came without pay...
Actually, if you look up the definition of aid in a dictionary or online, you can find in the list of definitions that you are given that "aid" can act as a noun meaning "a person or thing that aids or furnishes assistance; helper; auxiliary." Thus we see, that whilst to our minds it seems more natural to use the word "aide" in this context, at the same time, according to the dictionary, there is no problem with using the word "aid" in this context as it means the same thing. Let us remember after all that "aide" actually comes from the French word for "aid."