On Sept. 7, at about noon, Mr. Mengistu arrived at the family apartment in an agitated state. “He was pacing and wrapped up in his own thoughts,” said Ilan, another brother. His mother went into the kitchen to cook, hoping he would calm down. Then she heard the front door slam. By nightfall, Mr. Mengistu had not returned.

The next day, Ilan Mengistu said, officers from the Shin Bet internal security agency called him, then came to him at work and told him that his brother had crossed the border into Gaza at around 5 p.m. the day before. He left behind a bag containing his identity card, a Hebrew Bible, a towel, slippers and a mathematics book, though his brothers said he was not known to be studying math.

Mr. Mengistu had never spoken much about Gaza, according to his relatives. During last summer’s war, militants fired thousands of rockets at Israel, more than 2,200 Palestinians were killed in Gaza and more than 70 were killed on the Israeli side, mostly soldiers. But Mr. Mengistu was “in his own world,” Ilan said, adding, “What was happening did not interest him.”

When Israel publicized Mr. Mengistu’s case, it said another Israeli citizen, a Bedouin Arab from the Negev desert who was not named, had also crossed into Gaza in recent months and was also believed to be in the hands of Hamas. Israel says Hamas is also holding the remains of two soldiers, Second Lt. Hadar Goldin and Staff Sgt. Oron Shaul, who were killed in action last summer.

The government has paid a high price in the past for the return of its citizens and soldiers, or their remains, in lopsided prisoner exchange deals. But the issue has become more politically charged since 2011 when Israel released more than 1,000 Palestinian prisoners, many of whom had been convicted of deadly attacks against Israelis, in exchange for Gilad Shalit, a soldier who was seized in a cross-border raid in 2006 and held captive by Hamas in Gaza for five years.

Gathered in Ilan’s small and simply furnished apartment for a recent interview, Mr. Mengistu’s family now seems torn between treading carefully with the Israeli authorities and not wanting to drive up Mr. Mengistu’s price, while appealing for mercy from Hamas and for international support.

In a rare public reference to Mr. Mengistu, a senior Hamas official, Moussa Abu Marzouq, denied Israel’s version of events, telling Al Jazeera’s website in a mid-July interview that Mr. Mengistu — who never served in the army — had been photographed in military uniform and suggesting that he was already in Gaza during the war last summer. He gave no hint of Mr. Mengistu’s whereabouts or condition, but he claimed that the Israelis were pretending that Mr. Mengistu was insane in an effort to lower his value.