Here's my two cents, for what it's worth.
In my experience, it can be most productive to start with your grandmother and any records that are connected to her. These might include (if they exist): birth certificate(s); adoption papers; baptism record(s); adoptive/foster family recollections; marriage record; death record; etc. If you have her date of birth, you could search for a birth notice in the local newspaper.
In terms of searching for this Alexander Brown, have you searched for an obit for him in the local newspapers? Have you sent queries to the local cemeteries and funeral homes? Have you asked the local libraries if he appears in any city directories (which might list his home address and possibly his wife)? Since you know he died in 1920, it's possible that his widow would have been in the city directory at the same address in 1921 or 1922.
Good luck in your search.
