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The Wiener index is a topological index of a molecular graph, defined as the sum of distances between all pairs of its vertices. Benzenoid graphs include molecular graphs of polycyclic aromatic hydrocarbons. An edge thorny graph <i>G</i> is constructed from a catacondensed benzenoid graph <i>H</i> by attaching new graphs to edges of a perfect matching of <i>H</i>. A formula for the Wiener index of <i>G</i> is derived. The index of the resulting graph does not contain distance characteristics of elements of <i>H</i> and depends on the Wiener index of <i>H</i> and distance properties of the attached graphs.