Set Cover with bounded intersection size
If I'm not missing something, you can use a reduction from SINGLE OVERLAP RESTRICTED EXACT COVER BY 3 SETS (SINGLE OVERLAP RX3C) which I proved to be NPC in this cstheory question.

EXACT COVER BY THREE SETS (X3C):

Instance: Set $X=\{x_1,x_2,...,x_{3q}\}$ and a collection $C=\{C_1,...,C_m\}$ of 3-element subsets of $X$.
Question: Does C contain an exact cover for $X$, i.e. a subcollection $C′\subseteq C$ such that every element of $X$ occurs in exactly one member of $C′$?

X3C is NP-complete (see G&amp;J), and as shown in [1] it remains NP-complete even if every element $x_i$ is contained in exactly 3 subsets of $C$ (RESTRICTED EXACT COVER BY THREE SETS, RX3C).

I proved that it remains NP-complete even if every pair of subsets in $C$  share at most one element; i.e. for all $i\neq j$, $|C_i \cap C_j|\leq 1$ (I called this restricted version SINGLE OVERLAP RX3C).

The SET COVER WITH BOUNDED INTERSECTION SIZE 1 (its decision version)  is simply a generalization of SINGLE OVERLAP RX3C, indeed you can pick the same universe $X$  and the same collection of subsets $C_1,...C_m$ of the SINGLE OVERLAP RX3C and ask for a cover with $q$ subsets or less.

Obviously a cover with $&lt; q$ subsets cannot exist because every subset contains three elements and there are $3q$ elements in $X$. A cover with $q$ subsets must be exact: if two subsets contain a shared element then there are less than $3q$ elements covered.

[1] Teofilo F. Gonzalez: Clustering to Minimize the Maximum Intercluster Distance. Theor. Comput. Sci. 38: 293-306 (1985).