Islamic Jihad, al-Aqsa Martyrs Brigades and a previously unknown group calling itself the Army of Believers each said they carried out the attack.
Khaled al-Batsh, a senior Islamic Jihad leader, said the attack was "a natural response to the continued crimes by the Zionist enemy".
Islamic Jihad is not one of the Palestinian actors subject to a Gaza ceasefire agreed upon in November between Israel and some Palestinian groups.
The group has demanded that the truce also cover the occupied West Bank.
Bruno Stein, Eilat's police commander, said the police believed there could be more bombers in Eilat.
"Our assumption is that it's not one bomber, and there might be more bombers in Eilat right now," he said.
The attack is the first suicide bombing in Israel since last April when a suicide bomber killed 10 people in Tel Aviv.