Neoptolemus was the son of Achilles by Deïdamia. His name means 'young warrior', from neos, 'young' and ptolemos, Epic Greek for polemos, 'war'. (The latter word is the origin both of the name of the Ptolemies, the Greek kings of Egypt, and of English 'polemic'.)
This was the name given to him by Phoenix, son of Amyntor, either because of his youth or, more likely, because he joined the army late in the war against Troy, i.e. most of the soldiers were old stagers, while he was a new recruit. His name before this was Pyrrhus (which see).