Huck has faked his own death. This is the simple explanation behind many of his lies.

If Huck tells people his real name, the fact that he is still alive may be discovered. This would be a problem for Huck because he believes that his father will come after him and renew his demands to 1) raise Huck and 2) acquire Huck's money.

Huck wants to be free of his father and so continues to "cover his trail" by lying. Later he learns that Pap Finn is dead, but for the entirety of the action of the novel Huck works under the presumption that Pap is alive and represents a danger.