In a statement, the family said "given the circumstances of Aysenur's killing, an Israeli investigation is not adequate."
"A US citizen, Aysenur was peacefully standing for justice when she was killed by a bullet that video shows came from an Israeli military shooter," the family said. It urged the US to ensure accountability "for the guilty parties."
The family said Eygi had graduated from the University of Washington in Seattle three months ago.
"Aysenur felt compelled to travel to the West Bank to stand in solidarity with Palestinian civilians who continue to endure ongoing repression and violence."
Palestinian and Turkish officials said Israeli troops shot and killed the 26-year-old.
The White House has said it was "deeply disturbed" by Eygi's death, but did not say whether she had been shot by Israeli troops. It has urged Israel to investigate.