Since the information you have so far is not helping, it might be better to try an indirect approach. Here are some things to try.
Make a list of all the source material you have about your grandmother's life, and then go through the documents systematically. Make a list of events and put them in order on a timeline. If your grandmother has siblings, do the same thing, and make a family timeline. Work backwards slowly to see if you can locate what part of New York the family might have been in when your grandmother was born.
The FamilySearch Wiki's article New York State Genealogy has links to articles on Research Strategies and to a Record Finder. You can use these to look for other records that might help you narrow down the search area, such as baptism records. But you may have to widen your search, look at more record types, and establish the migration path of the entire family in order to solve this problem.
Keep in mind that places change names, and sometimes records give as a birthplace the name of a larger populated place which is close by. Using the USGS's Geographic Names Information System (GNIS), I searched for populated places including the term 'West' and got 242 results. The results show which USGS map the places are shown on -- you can look at these maps online using the Historical Map Explorer.
Another possibility is that the place referred to is not the name of a populated place, but it is a neighborhood within a larger place. Consult a reference guide such as the NYG&B's NY Family History Research Guide. You can find this in their shop or in a library near you, using WorldCat. Or you could look for a gazetteer, a dictionary of places, using Google Books, the Internet Archive, Hathi Trust, or WorldCat.
Another complication is that your grandmother may not have been born in the area where your family lived, so if you can find extended family members, map out where they were living as well.
The date of the application for the Social Security number is an important clue, since it tells you how likely the birth place is wrong or only partially correct, given the amount of time which had passed between the birth and the application. Do you have the SS-5 itself, or are you only looking at Ancestry's Social Security Application and Claims index?
Bear in mind, too, that people can't self-report their birthplace. We have to take the word of someone else about where we were born, so the information on any record initiated by your grandmother is second-hand information.
As a wild guess -- could your grandmother have been born somewhere in Westchester County?
If you still can't make any progress after widening the search, consider hiring a professional. Many associations of professional genealogists have directories that will help you locate a professional who specializes in New York research, such as: