I found that after changing the nickname, it might take a short while before the nickname will change across Google.  I also found that closing the browser and re-opening was sometimes necessary to see the new nickname.  But in the end it worked, and I successfully changed my nickname on my custom Google Maps.

From what I understand, this also changes your display name on Google Sketchup, Google Docs, and perhaps other Google services, so keep that in mind before you change it.