That is an interesting question. I think it did have to do with her name. The show didn't spell it out, but I think it could make sense with your scenario, or the following one:
The storybook did say Snow and Charming's name was Emma and that she would be the Savior. I don't think Henry would have immediately connected himself with the Savior.
Maybe completely separately, Henry was desperate to escape Storybrooke (because the book made him believe the idea of the Curse and was freaked out by everyone else aging except himself), so he stole MM's credit card to find out his birthmother's name.
I think he connected the dots when he found out that his birth mother was named Emma, and thus concluded that she was the Savior from the storybook.