There is not a single store that gives refunds on "Gift Cards".  But you haven't lost the money..you can still use it to purchase items at Target.

You infer that if they don't refund the card they will "loose" a customer.  So that means that if they refund it you will stay a customer?  If that is the case then what the heck is the problem.  That is what does it matter if you use cash in the future to purchase something or this gift card?  So just use the card on something you would have bought as a customer in the future and move on..there are more important things in life to worry about.

Now, I will say that since it appears you live in California, by law they(or any store) must give you the cash back from gift cards in certain circumstances.  First the card must be a "Store Card" and not one that can be used at many stores(this should be your case).  Next, there must be $10 or less left on the card.  If this applies to you then you can take it to Target and they will give you the cash back from the card..and yes I have done this several times.