How do I change what other people see on their caller id when I call them?

What do you mean "change it"? It has to be the name on the account the phone number is on(or atleast a authorized user?). If your looking for Tom instead of Tommy or the spelling is wrong, then thats one thing, but it can't be custom. Thats only available to Cox Business customers.

I had to do this with my Verizon cell phone. It just said Cell Phone on the caller ID box.  Verizon re-routed me to a service ( sounded like Luminate or Illuminate).  They asked me what name i wanted to appear and then asked me to say into the phone that i owned the account and requested the change.  It worked. 

Cox must be able to fix your name.  Call their support number and push the button for Phone Support.  If people don't see your name on their Caller ID, they will think it is a telemarketer and not answer.