ABSTRACT We study the build-up of the stellar mass of brightest cluster galaxies (BCGs) at z < 0.1 as a function of their morphological type. Morphometric parameters for 1216 BCGs were obtained from the SDSS DR12 r-band images with the code morfometryka. We show that the morphometric parameters are efficient to separate BCGs into cD galaxies, conventional ellipticals, disc-dominated BCGs, and merging objects. We propose a simple classification scheme using two concentration indices, C1 and C2, and the spirality σψ. We then investigate how the stellar masses of BCGs and the velocity dispersions of their parent haloes relate to their morphological properties. We find that BCGs with larger envelope contributions are more massive than BCGs in other classes and more common in massive clusters. The stellar mass and the halo velocity dispersion correlate for all classes, suggesting that environmental effects are important for BCGs of any morphology. At fixed parent halo mass, we detect a continuous mass excess up to $\sim 60{{\ \rm per\ cent}}$ from pure E galaxies to extreme cD galaxies, while the inner regions of such cDs present only $\sim 40{{\ \rm per\ cent}}$ of the mass of an E galaxy. We interpret our findings in a scenario where different BCGs have developed their mass through an asymmetric combination of distinct processes, so that present-time cD galaxies have grown in mass mainly through minor merging and tidal stripping from satellite galaxies, while elliptical BCGs evolve mainly through processes like major merging.