Is this a well-known NP-hard problem?
3-Hitting Set problem is known in parameterized complexity theory. The requirement $\cup S_i=R$ can always be assumed without loss of generality. See e.g. An efficient fixed-parameter algorithm for 3-Hitting Set. According to this link it is NP-hard in its usual (not parameterized) form. Proving NP-completeness of your problem we give reduction FROM 3-Hitting Set to your problem not vica versa. Therefore your problem is NP-complete (in its decision form).