I hear this most frequently in reference to two types of names:
- Female names -- especially non-traditional ones -- that end with a long-e sound (e.g. Kinsley, Shaylee, Rylie)
- Male and female noun-names that are either edgy or cutesy (e.g., Blade, Blaze; Precious, Essence)
This is a tricky topic, because it's so subjective. Names that end with an "ee"-sound might seem adorable to some, cloying to others. Names like "Blade" and "Precious" might seem strong and/or youthful to one person, strange and/or silly to another.
One the one hand, many parents have told me that they're children "grew" into their names as they got older -- that names just seemed to fit after enough time. So perhaps those who don't think a name can "age" just haven't given certain names a chance.
On the other hand, though, society at large isn't as willing to get used to particular given names. For instance, I think most people would rather hire a lawyer named Stanley than one named Blaze, or see a dentist named Lauren over one named Shaylee.
So when you're trying to decide if a name will "age well," you've got to consider both your own feelings (as a parent) and also the way other people will perceive the name in a variety of possible scenarios.
Also -- just to make things even more complicated :) -- you've got to keep in mind that our impressions of names change drastically over time. (Florence and Walter were very popular names a century ago, but they're basically shunned today.) So people who can't get behind the (fairly new) name Nevaeh just yet may only need a decade or so to become accustomed to it and change their opinion of it.
So I guess I can't quite answer the question of whether certain names do or do not "age well." In fact, it may ultimately be an unanswerable question (...unless, of course, the name you're considering is something like "Bugaboo" or "Poopsie-Bear"). But it's definitely something to mull over if some of the names you're considering fall into one of the above categories (i.e., edgy, cutesy, or a long-ee ending).

