Ok, the good news first (this is as of April 17, 2013). You can add a credit card to your account online and you should give it a nickname. Go to straighttalk.com, log in, select Manage Credit Card information on left side, select Add New Payment Method and carefully add the new credit card. Don't look for an online option to assign Auto-Refill to this new card, there isn't one. Also, the online chat service for Straight Talk won't make this change for you either.
Now the not so good news. You have to call the dreaded voicemail tree. Here is the sequence:
call (877) 430-2355
select 1 for english
select 5 other [don't bite on the automated refill option, this won't help]
select 6 other
select 2 other
select 1 for active phone
enter your straight talk phone number
enter 1 for yes
wait for a human being, and explain that you added a new card online, and you want customer service to switch auto-refill to it. The English quality on this line is not so good, so I prefer not to give the card info orally and have it transcribed wrong.