to stay somewhere for a stretch of time: to settle into a place for days or weeks instead of passing through
Luke writes like a travel narrator, and this is one of his markers for a real stop. Set against the bare verbs for arriving and sailing on, this word says the travelers did not simply pass through: they settled in and put in time. Twice he names what filled that time, teaching and preaching the word of the Lord at Antioch, or speaking boldly at Iconium, so the length of a stay doubles as a measure of the work done there.
The one verb tilts two ways. Sometimes the weight falls on the place: Jesus went into the Judean countryside and stayed there with his disciples (John 3:22), and Herod went down to Caesarea and stayed (Acts 12:19). Sometimes it falls on the clock: seven days at Troas, some days at Philippi, eight or ten days before a hearing, and once simply a long while. Greek holds both with the same verb, while English usually reaches for "stayed" in the first case and "spent" in the second.
It sits close to the ordinary verb for remaining (μένω), but it is narrower. That general verb can mean to stay anywhere, to keep on being something, or, in John, to stay joined to a person. This one keeps its feet on the ground: a person, a place, and a stretch of days.