I don't have access to Minecraft on Linux at the moment, but that command line looks suspicious. Give this a try:
java -Xms512m -Xmx1024m -cp "$HOME/.minecraft/bin/*" -Djava.library.path="$HOME/.minecraft/bin/natives" net.minecraft.client.Minecraft "$USER"
If it still isn't working for you, knowing what does happen (error messages, lets you in but has wrong name, …) would be helpful for further troubleshooting. :-)
Edited: Work on Ubuntu 12.04 x32
*P.S.: after "$USER" you can also add "$PASSWORD" "$SERVER" *
You can do it, but not through the Minecraft launcher. Open TextEdit and make sure it's in plain text mode. If it isn't, choose "Make Plain Text" from the Format menu. Then enter the following text:
#!/bin/bash
cd ~/Library/Application\ Support/minecraft
java -Xms512m -Xmx1024m -cp "bin/*" -Djava.library.path="bin/natives" net.minecraft.client.Minecraft "YOURNAMEHERE"
Replace YOURNAMEHERE with the name you want, but keep it in quotes. Now save your file, and make sure you uncheck the box that says "If no extension is provided, use '.txt'." Save the file as launcher.command (you can call it whatever you wish, just make sure it ends with .command).
Now in order to make this launchable with a double-click, open up Terminal and type sudo chmod 777 and then drag and drop your file on to the Terminal window. Then press Return. It will prompt you for your password, which you should enter. You can now close Terminal.
The file will now be executable, so you can run it by double-clicking. It will open Minecraft in offline mode with the custom name. If you ever want to change the name, right-click the file and choose Open With > TextEdit. You can then edit the name and save the file, and it will use the new name.
Best Answer
Open up a notepad file, and put this command into it.
Replace
yourusernamewith whatever name you wish. Save the notepad file asCUSTOMNAME.batYou can use that file to now run minecraft with your name changed. (you may have to go into your folder options and enable the viewing of system files to see the customname.bat after you have saved it depending on your current windows settings)