I have heard people say things like "don't wake him up that's gezel shainah". I would like to know if that phrase has any basis in halacha and if so what are the parameters? For example if I wake someone up for minyan is that a mitzvah haba bi'dei aveira?
Short answer: no, it's not a formal prohibition, but it's still wrong to prevent someone from sleeping.
The sefer “Ve-Ahavta Le-Re’acha Kamocha” notes that Rav Chaim of Brisk used the expression “gezel sheina,” implying that waking somebody up needlessly is a form of theft.
However, there does not appear to be any grounds for connecting waking somebody up with monetary theft.
It is true that there are other forms of theft that do not involve monetary loss, such as geneivas daas, which the Tosesfta writes is the most severe of all types of theft. However, even the deception of geneivas daas involves taking something from the victim — his daas — whereas preventing somebody from sleeping is a physical nuisance, but doesn’t transfer anything from the victim to the would-be-thief.
This point has been raised by Rav Shmuel Wosner (Shevet Ha-Levi 7:224), who writes that denying the benefit of sleep from a person is certainly a sinful act (see Bava Basra 20b; Shulchan Aruch, Choshen Mishpat 156, concerning the laws of neighbors), but does not fall under the category of theft.
According to HaRav Aviner it falls under "Love your fellow as yourself"
Q: I work as a ticket checker on the bus to insure that all the passengers paid. Is it permissible for me to wake someone up or is it forbidden on account of "Gezel Sheina" – stealing someone's sleep?
A: You are obligated to wake him up. "Gezel Sheina" is forbidden on account of "Love your fellow as yourself", but here it is necessary (See Piskei Shlomo Vol. 3 p. 142).
And is therefore not considered actual theft
The Mishneh Halochos has a Teshuva in Shu"t Mishneh Halachot Vol. 12 Sec. 443. He first goes into an explanation from Brachot how its a bad thing to do, then brings a source from Sanhedrin that on the contrary it's possibly a Mitzva to wake someone.
In conclusion it applies to someone who is sickly or hasn't slept for three days.
Based on this it would be permissible to wake a [healthy] person for a Dvar Mitzva.
Rashi Avodah Zarah 24a writes that depriving someone of sleep is called being metza'er him.