Catalyzes the NAD-dependent oxidative cleavage of spermidine and the subsequent transfer of the butylamine moiety of spermidine to the epsilon-amino group of a specific lysine residue of the eIF-5A precursor protein to form the intermediate deoxyhypusine residue. [eIF5A protein]-L-lysine + spermidine = [eIF5A protein]-deoxyhypusine + propane-1,3-diamine Protein modification; eIF5A hypusination. Belongs to the deoxyhypusine synthase family.