We present lithium observations of 23 X-ray selected candidate members of alpha Per, which are part of a larger sample of stars identified through two ROSAT surveys of the cluster. Our observations on one hand allowed us to confirm membership for 18 of the candidates, thus suggesting that a high percentage of the whole X-ray selected candidates are probably cluster members. On the other hand, we had the possibility to significantly enlarge the Li database for this cluster. The distribution of Li abundances for stars in our sample (or `new' members) is in good agreement with that for previously known (or `old') members, although `new' members in the 5000 - 4700 T_eff interval stay on the upper envelope of the Li vs. T_eff diagram. The comparison of the merged `new' + `old' sample with the younger IC 2602 and IC 4665 clusters and with the older Pleiades confirms that stars more massive than the Sun do not undergo any PMS Li destruction, whereas some depletion occurs during the early phases on the ZAMS. We re-addressed the issue of the star-to-star scatter and Li-rotation connection for both alpha Per and the Pleiades; as several previous studies have pointed out, fast rotators, as a group, show higher lithium than slow rotators. At the same time, however, fast rotators exhibit a much narrower dispersion than slow rotators. We demonstrate that this dicothomy is unlikely due to projection effects and suggest that the reason for it could reside in the PMS rotational history and, in particular, in the presence (absence) of a circumstellar disk. As to very cool stars (T_eff < 4500 K), we find that alpha Per members do not seem to have higher lithium than the Pleiades. This result, however, must be confirmed with a larger sample of alpha Per stars before any conclusion can be drawn. Based on observations made with the Isaac Newton telescope, operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof\'i sica de Canarias.