The Israeli army said soldiers spotted the woman approaching them with an explosive device near the refugee camp and threw a stun grenade at her. The woman detonated the device, killing herself, the army said.
"I offer myself as a sacrifice to God and to the homeland"
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Al-Nejar's family said she had nine children and nearly 30 grandchildren.
Nour Odeh, Al Jazeera's correspondent in Gaza, spoke to al-Nejar's family who said that they believed she had decided to take action because of the suffering she had seen of the Palestinian people throughout her life.
"The family told me that she simply couldn't take it any more," Odeh reported.
"Six out of seven of her children were imprisoned and injured by the Israeli army. Her house was demolished and her grandson was shot dead by the Israeli army in 2002."
Al-Nejar was the first known Palestinian grandmother to attempt a suicide bombing against Israelis.
Fouad, one of her sons, 31, told Reuters: "I am very proud of what she did. Allahu akbar [God is greatest]."
On a video released by Hamas, the woman read out a statement saying she wanted to dedicate her death to the Palestinian prisoners held in Israeli jails and to Ismail Haniya, the prime minister. Thousands of Palestinians are being held by Israel.
In the video she wore a black explosives belt and had an M-16 assault rifle slung over her neck.
"I offer myself as a sacrifice to God and to the homeland," she said.