Should one sit shiva for a non-Jewish relative? Say your mother is Jewish and your father is not, and he dies.
Because the parent is not Jewish, there is no chiyuv (requirement) for you to sit shiva for them.
That being said, there is a kibud av v'em (honor for father and mother) issue which must be contemplated when dealing with sitting shiva for a non-Jewish parent. Basically, there is a concern that the person would not be showing proper respect to his deceased parent by not sitting shiva. As kibud av v'em is one of the five things that a person is rewarded both in olam haba as well as olam hazeh (this world and the next), it is an important concern to contemplate.
Furthermore, if a non-Jew was to ask that person why he wasn't sitting shiva, observing shloshim or saying kaddish and he was to respond that his parent was not Jewish, this could lead to a chillul Hashem (desecration of the name of G-d) which could be detrimental to the Jewish people.
(This is more important when discussing gerim [converts]; if someone converts, halachically their birth parents are no longer their parents, as Avraham Avinu and Sarah take their place. However, resentment can arise if someone was to ask why they weren't sitting shiva and the response is "they're not Jewish" or "under Jewish law, they aren't really my parents anymore.")
So, there is room to work here. I would say that, in the case of a non-Jewish parent that the child wants to mourn by sitting shivah, there is a sufficient case to be made for allowing it. However, for a child who does not want to mourn, there is no absolute requirement for them to mourn.
My answer to this would be the same as I wrote to an earlier question about whether a convert can say kaddish for his birth-parents, with a minor caveat since you're speaking about an individual who has one parent who is Jewish and another who is not. In the case where the mother is Jewish, there is no question that the child says kaddish and sits shiva for his or her mother since the child is Jewish by birth (assuming that the mother is no longer with the non-Jew and not worshipping in a non-Jewish forum).
But in the case where the child's father was Jewish and the mother would not, the child would be considered non-Jewish, according to Jewish law, until she or he converted. Upon conversion, the convert is normally considered like a newborn child and the birth-parents are generally discounted. Babyl. Talmud Yevamos 97a-98b. Maimonedes (aka Rambam) holds that conversion breaks blood lines for purposes of mourning. Avel 2:3. So even if a father and his daughter both converted together, she would not sit shiva for him. Ibid. Although I haven't found a ruling for the case of a convert with a Jewish father, I would think it would be much the same things. The conversion has technically broken the father-daughter tie and she would honor her father in other ways, such as reading Psalms, or dedicating time to study Torah in his memory.
As I wrote before, my Rav told me that I could not sit shiva for my parents. The reasoning is that someone may assume -- based on the convert sitting shiva -- that the deceased parents were Jewish, and then make false assumptions about whether siblings are Jewish, also, and might consider introducing the non-Jewish sibling to a Jewish single for purposes of possibly getting married. This is a more serious possibility than saying kaddish for a non-Jewish parent because no one would necessarily know for whom you are saying kaddish. If a convert cannot sit shiva for a non-Jewish parent because someone might assume the parent was Jewish, then I think it is obvious that a child of a mixed marriage cannot sit shiva for her non-Jewish father.