My name was misspelled in the official records, which my parents didn’t learn until I was nine, and they applied for a passport for me, so it’s misspelled on my first passport. The final “H” is missing from my first name.
I managed to get my driver’s license issued with the H though, because I could show them my high school enrollment, and I had been enrolled as “Rivkah” all through school, except for a brief experiment with calling myself Rebecca, and then I got my high school diploma and college degree with “Rivkah.”
So when I entered the military, I enlisted as Rivkah, and went all through training as Rivkah, had dogtags issued as Rivkah, military ID, etc., and went all through training as Rivkah.
When I got to my home unit, some pedantic paper pusher (newly hired and out to prove something) decided that my name had to be what was on my birth certificate, and set out to change it on all my records. That was going to be a real headache for me, because now it wouldn’t agree with my college transcript or civilian driver’s license.
So I looked into having it fixed.
I was told (state of New York) that all I needed to do was show that I had “always” used the Rivkah spelling, indicating that it was indeed a mistake, and not a recent personal preference, which was easy, because among other things, I still had birth announcements my parents had made up when I was born with the “H” spelling. I sent one, along with a copy of a first grade report card, a cancelled receipt from my paper route when I was 11, my high school diploma, my driver’s license, my college transcript, and several other documents, showing a lifetime of using the spelling. The record was corrected for free, but I did have to pay $10 for the new copy of my birth certificate.
I imagine that your state is similar. All you need is clear evidence of the error-- in this case, a copy of your wife’s own birth certificate, and maybe something else, like a copy of the birth announcement from the paper, or the hospital discharge, if you saved stuff like that as mementos. They should correct the record, and then you will probably have to pay for a copy of the new certificate, but eh, it probably won’t be much.
I’d do it. You never know, with tightening security in this country, the possibility of national IDs, and like you say, what if she ever wants to run for office and is expected to release her birth certificate? It could be a problem. It could even be a problem if some employer decides it must be fake, and not to hire her. You just never know. If it’s an employer who has been burned by fake IDs before.
Start by writing or calling the office that issues BCs in whatever county she was born, and see what the procedure is.