Endowing room-temperature ductility in refractory high-entropy alloys (RHEAs) is a challenge to their uses in nuclear energy systems, biomedical, and high-temperature applications. Recently, transformation-induced plasticity (TRIP) has been recognized as an effective strategy to simultaneously improve ductility and tensile strength of RHEAs. Hitherto, the design for a TRIP mechanism in RHEAs through material-dependent parameters typically follows empirical approaches. Here, we investigate the alloying effect of several body-centered cubic (bcc) transition metal elements (TM=V, Nb, Cr, Mo, and W) on the phase stability and the micromechanical properties of the TiZrHf alloy using a first-principles method. We show that the addition of the considered TM elements increases the stability of the bcc phase relative to the hexagonal close-packed (hcp) phase and the relative stability between these two phases can be tuned and inverted. We investigate the composition-dependent single-crystal elastic constants for the (TiZrHf)1-xNbx and (TiZrHf)1-xMox alloys and analyze mechanical stability, elastic anisotropy, and polycrystalline moduli. Our results show that the anisotropy of Young's modulus is more pronounced the closer the alloy composition is to the composition where the bcc phase or hcp phase becomes mechanically unstable. We find that the hcp phase has higher shear and Young's moduli than the bcc phase below a critical composition for the Nb or Mo addition, while the bcc phase has larger moduli above the critical composition. Furthermore, our results imply that the d-band filling has a dominant influence on the phase stability and mechanical properties of the alloys.