The name is shown in several places and they can all be different.
1) when downloading a map from some one you can see the name of the file itself. so if you rename the map to "lolz you see download now" thats what he will see at the top right UNTILL HE FINISHES THE DOWNLOAD!
2) Game Lobby and Map selection screen (map selection screen is where you choose what map you want to host from your map list)
To change those names you need the following tools
a) hex-editor (like XVI32)
b) MPQ-Editor (like MPQMaster)
c) some text editor (like jasscraft)
open map using ur MPQEditor and extract war3map.j
open war3map.j using your text editor
look for function config and under it look for
Now, if the map name is over there, change it. If it says TRIGSTR_X(some number not really X) so extract the file called war3map.wts from the map using MPQEditor. open it using your text editor and look for string X(x being the number from the TRIGSTR_X) there you will see the maps name.
Import the file you edited back to the map using your MPQEditor.
close the MPQEditor
Open the hex-editor and browse for your map
on the right side youll see the maps name. change it. NOTE: if there is something you want to delete (in the name) so click on it and press the space bar to make it empty DO NOT delete anything using the backspace.
Recommend - dont save over the same map as you could have messed something up and you cannot undo it.
Now the names have been changed.