What is the sources for the practice to stay up late learning torah on Thursday nights? When or how did this practice develop and spread? And why is it called a mishmar? Is this a common name for a seder limud?
"Why learn late thursday night?" To avoid a fast. The most religious among us love to fast from Thursday evening to Friday evening, in order (for them) to enjoy (more) the delight/the oneg of Shabbat celebration. Those among us, however, who are unable to fast or do not want to fast (especially from Thursday evening meal), arrange to learn Torah or a tractate of the Talmud (late into Thursday night), with the express idea of concluding with a seudat mitzvah, which takes precedence over a fast, or thereby avoiding a fast.
UPDATE (for the benefit of various comments): Some pious Jews also fast every Monday and Thursday in commemoration of the destruction of the Temple, of the burning of the Torah .... The first and second Mondays and the first Thursday of Iyyar and of Marḥeshwan, following the festivals of Passover and of Sukkot respectively, are recognized fasts in most Jewish communities, and were originally instituted to atone for the sins that might have been committed in the pursuit of pleasure during the holidays (Ḳid. 81a; Tos., s.v. "Soḳobo"; Oraḥ Ḥayyim, 492). .... many fast during all the ten penitential days (Oraḥ Ḥayyim, 581, 2, Isserles' gloss). Some pious Jews fast every Friday, so as to partake of the Sabbath meal with a hearty appetite (ib. 249, 3). http://www.jewishencyclopedia.com/articles/6033-fasting-and-fast-days