One of the many risks of booking through opaque third-party booking sites is that in addition to the fare rules for your ticket established by the carrier, you have an overlay of a contract with the third-party. Those third parties make their money on volume and not having to spend time on individual tickets. Thus, they can make it very hard to accomplish simple tasks.
Don't rely on what the third-party tells you on the phone. Read the contract conditions you agreed to and determine what, if anything, a name change will cost. Once you have that it hand, call back and simply ask to have done what the specific contract condition requires (if it is financially viable).
Not to berate OP, but for the benefit of others, saving passenger details and the like is also a risky proposition unless you triple check everything before hitting the "OK" button. There are dozens of stories such as OP here on FT. Some end just fine with air carrier staff either ignoring or making the change at the airport and some end very poorly.