There are several serious bugs in the saved locations functionality. I found that if you 'pin' a location (say a building) on the map (by pressing and holding) it will show the address. But when you 'save' it, the details of the address get 'lost' when looking at your search history. I ended up with multiple entries in my list titled 'Shinjuku, Tokyo, Japan' (which is the equivalent of, say, 'Manhatten, New York, USA'. Clicking each one still takes you to the correct location on the map, but they are impossible to tell apart in the list. Also then when using the 'directions' widget on the map and picking the destination, these saved locations appeared as longitude/latitude! I wonder if these issues are specific to Japan? I submitted feedback about this.
That being said, overall the app is quite impressive, and I'm sure it will improve.
TL;DR You can't edit the name of your starred locations from the iOS Google Maps app.