I think you are probably over-reacting on the popularity thing (in a nice way).
Two of my kids have very popular names, both times because we were slightly ahead of the curve (everyone copied us ;-) No seriously.).
It DOES bother me, if I am honest.
However, I grew up with an unusual name-oddly enough, now a very fashionable name-and I didn't love it. I have a much more common alternative name (long story) which I used more as a teenager because I was so fed up of being the only _ under 120 in the country.
I think, honestly, kids don't normally want to be different. Honestly. They want to be as like their peers as possible and they do not revel in having a slightly different name. As an adult, I don't MIND having an unusual name but it doesn't add anything. It says more about my parents than me.
I think Eve is a beautiful name and if you love it, please stick with it.
All that said, we've changed both my girls' names middle names slightly over the years (we wanted to honour relatives, long story) and it is straightforward to do it before 12 months, and only slightly less straightforward after 12 months. TBH with forenames, no one cares that much, they are pretty easy to change.