I have a Verizon Galaxy S3 with Jelly Bean on it.
What I want is to long-press the Home button and hear a beep to prompt me to provide a voice command.
I had it set up this way before I installed the official Verizon version of Jelly Bean. I had Cyanogenmod before.
I know that I can long-press the "Menu" soft-key to open up Google Now, but that's not what I want, especially since it doesn't even activate the voice search.
You could install an alternate launcher like Nova and assign the buttons to do different things. Alternatively you can flash a ROM that has the home button going to Google Search by default.
Galaxy S3 stock ROM has S-Voice on home key double press (do a double click like with a mouse), its not Google's app but its not so bad at all.
If you want to launch another app (like Google's Voice Search), you need to decompile, edit and recompile the framework.jar of your phone to add that function. I would suggest you ask about the change in the Cyanogenmod forum, since it should be fairly simple.
Stock roms hardly comes with enhanced options. IMO you should switch back to Cyanogen based ROMS or any other custom roms. I prefer/rely Cyano as I am using it past 3 years and has lots of revealed options which is good and bad as well. Currently I own a OnePlus One with cyanogen 11S. Alternatively you can root your phone and look for any such app that will help you in modifying the internal stuffs like JRummy Toolbox etc.