The name of the link is generally not useful if the link is opened in "launch" mode (where one program launches another via MathLink and attaches to it--for example when a .NET program launches the kernel). You might want to get the name if you open a link in listen mode, because in that case you need to tell the other program what link name to connect to.
The name of the link is dependent on the protocol used. Using the Shared Memory protocol on Windows, the name might be just a short string of meaningless characters, whereas with the TCP or TCPIP protocols it will include a port number and machine name, such as 1234@machine.domain.com.