Motivated by earlier experimental and theoretical studies on the potential of different nanostructures for drug delivery, the adsorption and electrical sensitivity of pristine, Al- and Ga-embedded (8,0) boron nitride nanotubes (BNNTs) to a rizatriptan (RZ) molecule are explored using dispersion-corrected density functional density calculations. The RZ molecule is shown to be weakly adsorbed on pristine BNNT, with a modest adsorption energy of −0.46 eV and only a minor impact on its electronic properties. The replacement of a B atom with an Al or Ga promotes the reactivity of the BNNT toward the RZ molecule, as evidenced by calculated adsorption energies and charge-transfer values. Once the RZ molecule is introduced onto the BNNTs, the band gap of the tubes narrows. However, the amount of reduction in band gap values of Al- or Ga- embedded BNNT is more pronounced than that of pristine tube. As solvent effects are included, the desorption times of RZ from the surface of BNNTs are expected to be shortened. The influence of surface charge on the adsorption process of RZ molecules, as well as curvature effects, is thoroughly investigated. Overall, the high sensitivity, reasonable adsorption energies, and fast desorption suggest that Al- and Ga-embedded BNNTs might be served as effective drug delivery systems for RZ.