Catalyzes the hydroxylation of the N(6)-(4-aminobutyl)-L-lysine intermediate produced by deoxyhypusine synthase/DHPS on a critical lysine of the eukaryotic translation initiation factor 5A/eIF-5A. This is the second step of the post-translational modification of that lysine into an unusual amino acid residue named hypusine (PubMed:16371467, PubMed:16533814, PubMed:19706422). Hypusination is unique to mature eIF-5A factor and is essential for its function (By similarity).