In this work, we explore the use of the Shapley value in ontology-mediated query answering (OMQA) and provide a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a FP/#P-hard dichotomy for SVC for ontology-mediated queries (T,q) composed of an ontology T formulated in the description logic ELHI-bot and a connected constant-free homomorphism-closed query q. We further strengthen the #P-hardness side of the dichotomy to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.