Is Clique completion for intersection models hard?
Good question. I hope this is not a homework problem.

I don't know if this problems exists in the literature, so let's call your problem Min Set Clique Completion Problem (MSCC). Here is a reduction from Hitting Set Problem to MSCC. Hitting Set problem is a well-known NP-complete problem.

Instance of Hitting Set problem is as follows : Let $U = \{1,2,\dots,n\}$ be the ground set. Let $T_1, T_2, \dots, T_l$ be $l$ subsets of $U$. Find a set $H \subseteq U$ that has at least one element from each set $T_i$. The goal is to minimize $|H|$.


We construct an instance of MSCC with $U&#39; = U \cup \{n+1\}$ and sets $S_1, S_2, \dots, S_{l+1}$ such that for $1 \leq i \leq l$, $S_i = T_i \cup \{n+1\}$ and $S_{l+1}$ is an empty set. (Edit : this is wrong as pointed out in the comment.)
We construct an instance of MSCC with $U&#39; = U$ and sets $S_1, S_2, \dots, S_{l+1}$ such that for $1 \leq i \leq l$, $S_i = T_i$ and $S_{l+1}$ is an empty set (Edit : Use this along with Karolina's fix in the comments).


Now if we run MSCC on this instance the solution $S_{l+1}$ returned by MSCC is precisely the hitting set of the sets $T_1, T_2, \dots, T_l$.