The personal name of the first-born son was often the name of his paternal grandfather. In the later periods, the name could be that of his father.
Greek personal names could be simple, composed of a noun or adjective, with or without a suffix. Plato, for example, means broad-shouldered, although Plato is a nickname (unlike Dio Chrysostom, it wasn't just added, but replaced the given personal name): the philosopher is thought to have been named after his grandfather, Aristocles [Aristo- 'best' + kles 'glory of/fame']. A better, but less familiar example is Plato's father, who was Ariston, which means "best."
Greek personal names could also be 2-item compounds of nouns, adjectives, verbs or adverbs, in 2, 3, 4 or more syllables. Often, there were 2 syllables in each section, or 4 total. You see this in Demo-kritos (combining a word that can be approximated in English by "the people" + "judge", in some form or another) and Alex-andros (defend/help + man). Other times there were 2 syllables in the second part of the name, and one in the first (or vice versa). You see this in Dem-archos (the people + rule [among other meanings, based on arche]) and Krat-ippos (power + horse). Turning these names into meaningful English is sometimes tricky.
The order of compounds could be switched (Aristo-nikos [best + victory] or Nik-aristos), with the notable exception of names that contained the name of a god. In such cases, the god came first. Dio-genes contains the god Zeus (the root is Dio-) plus a form of the Greek for "birth". Apollo-dotos contains the name of the god Apollo and some form of the word "gift". Theo-kles contains the Greek for "god" + "fame" or "glory of," as we see also in Herakles ("the glory of Hera") or Plato's real name.