NNN did not say whether it verified the authenticity of the statement.
The statement, a copy of which was published by NNA, said the attack was to avenge people killed in the three-month battle between the Lebanese army and Fatah al-Islam fighters in the Palestinian refugee camp of Nahr el-Bared near the northern city of Tripoli.
More than 400 people were killed, including 168 soldiers, in more than three months of fighting which ended in September 2007.
Abdeh is near the Nahr el-Bared camp.
Military officials have not identified the cause of the blast that killed the soldier.
A military statement identified the victim as Osama Ahmed Hassan, 24.
In another development, a senior Palestinian official said on Monday that a would-be suicide bomber who was shot dead on Saturday outside a Palestinian refugee camp in Lebanon, was carrying a false identity card.
The Lebanese army shot dead the man as he approached a checkpoint outside the Ain al-Hilwah refugee camp in south Lebanon wearing an explosives belt and carrying a hand grenade.
It identified him as Mahmud Yassin al-Ahmed, a 28-year-old Palestinian.
But Munir Maqdah, the Palestinian Liberation Organisation security chief in Lebanon, told the AFP news agency that he was carrying a forged Palestinian refugee identity card and that the UN refugee agency had no record of such a person.
"The man was not from Ain al-Hilwah. No Palestinian official recognised his body," Maqdah said.