The name Jonah in the Bible
Jonah is a son of Amittai and minor prophet who worked during the days of Jeroboam the Second (782-753 BC), just after Elisha and just before Amos and Hosea. God sends Jonah to Nineveh but he flees to Tarshish. During the journey, a storm compels Jonah's shipmates to throw him overboard and he is swallowed up by a great fish (which was not a whale; something that is made very clear by the Hebrew wording of the story).
In Matthew 12:40, Jesus compares His three day stay in the grave with Jonah's famous three-day stay in the great fish (Jonah in Greek is Ιωνας, Ionas).
Another connection between Jonah and Jesus occurs when the Pharisees state that 'no prophet arises out of Galilee' (John 7:52). They were wrong because 2 Kings 14:25 states that Jonah was from Gath-hepher, which is a town in Galilee, in between the Sea of Galilee and Mount Carmel on the Mediterranean coast.
The story of Jonah can be explained in all kinds of ways, but one of them is equating Jonah with Yahwism at large, which was preserved thanks to the temporary exile in Babylon. This exile is traditionally much lamented but it appears that king Hezekiah deliberately tried to bring it about, and that with the approval of the prophet Isaiah. From 2 Kings 20:12 (and Isaiah 39:1) we learn that Hezekiah was quite friendly with king Merodach-baladan of Babylon, who sent a delegation to see about his colleague king. Hezekiah proceeds to show the Babylonians his wealth the way Jonah diverts to Tarshish, a proverbial port of commerce where he boards a ship, which can only have been a commercial cargo vessel. Isaiah tells his king that the Babylonians will come back to take his wealth and people, and comforts him by stating that his sons will be high officials at the royal court of Babylon, which is how Yahwism was preserved during the stormy times of the Assyrian conquests (2 Kings 20:18).
Etymology of the name Jonah
There's something deeply peculiar about the name Jonah. Pretty much all sources derive it of the root יון, and render the name Dove. Jones' Dictionary of Old Testament Proper Names, however, makes a striking observation (or perhaps even an error). Jones suggests that the Hebrew word for dove comes from the verb ינה (yana), meaning to oppress, vex, do wrong:
Taking the name Jonah from this verb, it would interestedly mean Vexer, Wrongdoer.
Traditionally however, the name Jonah is understood to come from the root group יון, which consists of two identical but unused and therefore untranslatable roots:
Note the striking similarity between the word ינה (yana) and the name יונה (Jonah). The difference is only the letter waw, which, when used as a vowel, doesn't change the meaning all that much. By some twist of fate, a dove was known by a word that also means to vex. In practical life the dove wouldn't be known as a vexer, but in the story of Jonah, when it's by no means certain what the name Jonah means, any Hebrew audience would have understood the name Jonah as a pun.
For a meaning of the name Jonah, all the consulted sources go with Dove, but it also means Vexer.