Regel Yeshara, 10, cited in Minhag Yisrael Torah O.C. 155:15.
"Nittel Nacht" originate from. Some suggest that it comes from the Latin "Natale Dominus", meaning, "The birth of our God". Others surmise that the use of the word Nittel etymologically comes from the Hebrew "natal", meaning "to have been hanged". Nittel can also be construed as the Hebrew word for "being taken away". These latter interpretations of the word Nittel would seem to support an obscure theory that Jesus was actually killed on Christmas. Alternatively, it may simply be a derogatory nickname that was used to refer to Jesus in all circumstances. The more likely meaning of the term Nittel is that of an acronym for "Nolad Yeshu Tet L'tevet", meaning, "Jesus was born on the ninth of Tevet."
Nitei Gavriel Minhagei Nittel footnote 5:1
Refraining from Torah study is intended to serve as a sign of mourning. Mourners, including all Jews on Tisha B'av, are forbidden from studying Torah, as it is an activity which makes one happy. The "mourning" was for the rivers of Jewish blood that have been spilled throughout the ages, all perpetrated in the name of Jesus. Some Chassidic Rebbes went so far as to even refrain from sleep on Christmas Eve lest they dream about Torah.
Igrot Kodesh 7:23 cited in Shut Reiach Hasade 1:17
The Tzemach Tzedek of Lubavitch was once "caught" learning Torah on Nittel Nacht but quickly repented.
Bnei Yissaschar, Regel Yeshara, 10
Chassidic legend relates that wild dogs would visit those who "violated" the custom and studied Torah on Nittel Nacht
Taamei Haminhagim 500, Minhag Yisrael Torah O.C. 155:15
The most predominant and logical of explanations for the origin of Nittel Nacht customs, namely, the abandonment of Torah study and the closure of batei midrash on Christmas Eve was in fear of possible pogroms. It was common in the Middle-Ages for the Christians to unleash pogroms in honor of their holidays. By forbidding Torah study there was essentially no reason for anyone to leave their homes on Christmas Eve.
Shem Mishmuel Vol 2 p.75
Torah study at home was also not an option as it was quite rare for individuals to own their own books in those years. There were even some regions where the Jewish community was told to turn out their lights on Christmas Eve for their own safety. Some suggest that with Chanukah in close proximity to Christmas, the dreidel was a game invented in order to pass the time while sitting in one's home waiting out any fear of pogroms.
Emunat Itecha (Rabbi Moshe Wolfson)
The gematria of Nittel is the same as Chanuka alluding that proper observance of Chanuka can override the concerns of Nittel.
Miketz; heard from Rabbi Faivel Smiles. (The Gematriatic connection is a major strech. A.E.)
Sefer Haminhagim Chabad, Hayom Yom p.12
A mystical perspective to the prohibition on studying Torah on Nittel Night has it that by studying Torah one is contributing positive spiritual powers to the world.
Nitei Gavriel Minhagei Nittel footnote 5:4
It was believed inappropriate to make such contributions on a night of widespread idolatry during pogroms, and inappropriate merriment. There was also the concern that one's Torah study may go on to unwillingly serve as a merit for Jesus' soul, which was also undesired. This idea corresponds to the teaching that Torah study and prayer are said to give respite to the souls of all the wicked.
Nitei Gavriel Minhagei Nittel footnote 2:4
Some communities maintain that one is only to abstain from Torah study until midnight, while others did so until the next morning.
Nitei Gavriel Minhagei Nittel footnote 4:11, as advised by the Chatam Sofer
In other communities the custom was to go to sleep in the early evening and then wake up for a Torah study session after midnight. It would reflect badly in Heaven for Jews to sleep soundly throughout the night while the Gentiles were up in their places of worship praying to God.
Nitei Gavriel Minhagei Nittel footnote 5:1
There was also a custom to refrain from marital relations or going to the mikvah on Nittel Nacht.
Sefer Baal Shem Tov Vol. 2:43a
Conception on Nittel Nacht will breed an apostate
Emunat Itecha; Miketz.
Joseph could not study Torah that night, hence he went to do "his work" (v.11) The attempted seduction of Joseph by Mrs. Potiphar occurred on Nittel Nacht
Nitei Gavriel Minhagei Nittel 2:2,3
When exactly is the "halachic" Christmas Eve is actually subject to dispute as well. Some authorities recommend that the eve of December 25, as is Christmas in most of the Western World, is to be observed as Nittel Nacht, referred to as "Nittel Hakatan". Others suggest of course that the true Nittel Nacht is to correspond to the January 7 celebrations, the Christmas of the Greek Orthodox, referred to as "Nittel Hagadol". Common custom among most Nittel Nacht observers is to observe Nittel Nacht in accordance with the majority Christian community where one lives.
Nitei Gavriel Minhagei Nittel 6:1
Nittel Nacht customs were never established to be observed in the Land of Israel as Christianity had never been a dominant force there. It is also argued that the holiness of the Land of Israel overcomes any of the mystical concerns surrounding Christmas, especially those related to superstition and impurity.
https://en.wikipedia.org/wiki/Nittel_Nacht
Known in Hebrew as Leil HaNital (Hebrew: לֵיל הַנִיטָל or לֵיל הַנִיתָל). Other names include Blinde Nacht (Yiddish: בלינדע נאַכט, "Blind Night"), Vay Nacht (Woe Night), Goyim Nacht ("Gentiles' Night"), Tole Nacht ("Night of the Crucified One"), Yoyzls Nacht ("Jesus Night"), Finstere Nacht ("Dark Night"), and Moyredike Nacht ("Fearful Night").[1]