This question is worded ironically. Of course there is no gift to the Jews given by the Nazis on Rosh Hashanah, which is the Jewish calendrical New Year (the 1st of the month Tishri), nor on the following Day of Atonement, Yom Kippur (the 10th of Tishri). A parallel is presented, however, between the first 10 days of the Jewish New Year and the actions of the Nazis.
On Rosh Hashanah, sins of the preceding year are cast off and names are entered or not entered into God's Book of Life. This determines a person's fate about life and death in the upcoming year: if your name is entered, you will have salvation if you die in the coming year; if not entered, you will be cast into Sheol if you die. Yom Kippur presents a last chance to cast off or atone for your year's worth of sins: it is one last chance to have your name entered into God's Book of Life before the Book is sealed for the year along with its list of names, .
"All of creation bears witness to the Greatness of God!" In days gone by, Rosh Hashanah had dominated my life. I knew that my sins grieved the Almighty and so I pleaded for forgiveness ....
Night, Chapter 5 parallels this--in a nightmarish reversal--in that a "selection" is announced for shortly after Rosh Hashanah. This selection, when it quickly arrives, is a process whereby each prisoner is examined to determine if they are too weakened to work. Those who are too weakened are "selected," with their numbers written down, and kept from work on the day of a second "selection."
"Before you go into the next room, try to move your limbs, give yourself some color. Don't walk slowly, run! Run as if you had the devil at your heels!"
Those who make it through the second selection without their numbers being confirmed (or sealed), are given one more chance; their fate of death by crematorium is not yet sealed in the Nazi's list of death.
In a painfully ironic use of the word "gift," this then is the "gift" the Nazis give: a selection during the 10 days of New Year and atonement whereby the deaths of the weakened Jews are sealed in the Nazi's list of death.