What kind of data structure should be used for working with dynamic sets where elements follow "normal distribution" and why?
A hash function might not map the nearby elements to the same bucket. For example suppose the hash function is $h(x) = x^2 \mod p$ where $p$ is some prime number say $97$. Suppose $x$ follows normal distribution with values between $1$ to $1000$. And, suppose that most of the values are concentrated around $500$.
If you compute the hash values at $x = 499$, $x = 500$, and $x = 501$,
the hash function gives the following values: $h(499) = 2$, $h(500) = 31$, and $h(501) = 62$. Note that, although the input values are close to each other, their mapped values are not close to each other and they lie in completely different buckets.
Designing a good hash function that uniformly maps the values across the table, mostly depends on a particular use case. But, you can always make your hash function more complex by adding more terms to it, for example, by adding more terms of the form $x^k$ to it, etc. It will hopefully provide a uniform spread of values across the table.