It's not an entirely unusual idea. I bought a Winchester Model 12 that comes pretty close to my birth year. (Don't bother trying to guess that it's a flintlock; it's not.) I always wanted one and just happened to get an opportunity to buy it. The manufacture date sealed the deal. There was a similar aged rifle that was also available, but it was a caliber I wasn't interesting in getting.
Interesting idea. I should probably do this, as well as get one for the kids and grandkids birth years too.
If it's something that will be passed down, I don't even think it has to be extra special. My favorite shotguns are single shots my great grandfather had (late 1800's). They're not worth much, but knowing where they came from is priceless.