The minimal number of such Sidon sets is $\Theta(\sqrt N)$.

On the one hand, any Sidon set in $[0,N]$ has size at most $\sqrt N+O(\sqrt[4]N)$ (due to [Erdős and Turán](https://en.wikipedia.org/wiki/Sidon_sequence#Early_results)), hence the minimal number of Sidon sets that cover $[0,N]$ is at least $\sqrt N+O(\sqrt[4]N)$.

On the other hand, let $p$ be the least prime larger than $\sqrt{N/2}$, and let
$$S=\{2pk+(k^2\bmod p):0\le k<p\}$$
be the [Erdős–Turán Sidon set](https://en.wikipedia.org/wiki/Golomb_ruler#Erd%C5%91s%E2%80%93Tur%C3%A1n_construction) (= Golomb ruler). Then the shifted sets
$$(S+a)\cap[0,N],\qquad -p\le a<2p,$$
cover $[0,N]$, as any element of $[0,N]\subseteq[0,2p^2)$ can be written as $2pk+l$ where $k<p$ and $l<2p$, and the difference between $l$ and $k^2\bmod p$ is between $-p$ and $2p$. This gives a covering of $[0,N]$ by $3p\sim 3\sqrt{N/2}$ Sidon sets.