Are there any relationships between the given name Pompeius (like in Gnaeus Pompeius Magnus) and the city name Pompeii (the city thas was destroyed in a volcanic eruption? What do they mean?
Pompeius probably comes from the Oscan praenomen Pompo, which is the equivalent of the Latin Quintus, as its root is pompe - traceable to the Greek pente, i.e. "five".
The name of the city might have the same origin - quoting the English Wikipedia page, this could suggest that either the community consisted of five hamlets or, perhaps, it was settled by the gens Pompeia.
On the other hand the Parco Archeologico di Pompei gave in a Facebook post three possibilities. "All of them are potentially believable", but the origin from pompe is indeed said to be the most accredited hypothesis - the gens Pompeia was not mentioned though. Here are the other two possibilities:
i) According to the geographer Strabo of 60 BC, it might derive from the greek pempo ("to sent"), for its proximity to the Sarno River where goods arrived and departed;
ii) Maurus Servius Honoratus speculated in the 4th century that the toponym could trace back to a hypothetical foundation by Hercules, a pompa Herculis ("from Hercules' triumph") which would then become Pompeii in time;