The freighter that arrived near the island in season 4 was the Kahana.
The Kahana was sent by Charles Widmore and carried a team intended to reach the island and capture or kill Benjamin Linus. Its crew included the four people introduced in “Confirmed Dead”: Daniel Faraday, Miles Straume, Charlotte Staples Lewis, and Frank Lapidus. Naomi Dorrit had arrived earlier as part of the same expedition and served as the first warning that the ship’s mission was not a straightforward rescue operation.
The survivors initially assume the vessel may provide rescue, but the freighter’s crew is divided and increasingly violent. Michael Dawson and Walt’s former enemy, among others, become entangled with its operations, while Martin Keamy leads the mercenary force sent to attack Ben and the island’s inhabitants.
The Kahana is not the Black Rock, which is an older sailing ship connected to the island’s history. The Searcher is the vessel later used in the rescue operation, while Elizabeth is Desmond Hume’s sailboat.