Is every ultrafilter generated by a small set?

In fact it is a more tricky problem to get a model with a "small-generated" non-principal uf over $\omega$. See Kunen p 289 (old book) p 345 (new book), where an uf base of cardinality $\aleph_1$ in a model with $\mathfrak c >\aleph_1$ is generically defined. Or Baumgartner/Laver in Annals of Mathematical Logic 17 (1979) 271-288 where it is shown that some ufs in $L$ remain uf bases in $\omega_2$-long Sacks-forcing iterated extensions of $L$.

Let $F \subseteq [\kappa]^{\kappa}$ be an independent family (every finite boolean combination has size $\kappa$) of size $2^{\kappa}$. Put $E = \{\kappa \setminus \bigcap A: A \in [F]^{\aleph_0}\}$. Let $U$ be an ultrafilter containing $F \cup E$. Then $U$ is not generated by fewer than $2^{\kappa}$ sets. You can easily generalize this construction to get $2^{2^{\kappa}}$ such ultrafilters.

It is, however, unknown if it is consistent to have a uniform ultrafilter on $\omega_1$ which can be generated by fewer than $2^{\omega_1}$ sets.