Which is more important for a girl: davening or kibbud aim? How do we know the order of the mitzvos we should be doing?
Often, the order of mitzvos will be determined by the classification of the mitzvah as a Torah or rabbinic mitzvah, and by other halachic factors.
In this case, although honoring one’s mother is a Torah mitzvah, which would take precedence over the rabbinic mitzvah of prayer, the case of honoring parents is special, for as the Talmud states, both parents and children are obligated in honoring Hashem, and performing His mitzvos. Therefore when a parent instructs a child to transgress a prohibition, the child must not listen to his parent, and must perform the mitzvah even if it offends the parent.
Yet, for a girl, the basic obligation to pray is fulfilled by reciting a number of blessings, such as the morning blessings over the Torah. Further to this, the full order of prayer would be waived by the mitzvah of honoring parents, though if this causes you to miss Shacharis, you should make it up at Minchah. Usually, the clash between honoring parents and prayer can be avoided by praying at a different time, or by explaining that one wishes to pray, for a number of minutes alone, and asking if this is okay.
There is a dispute among authorities as to a woman’s obligation in prayer. Some write that she is obligated to the same extent as men; some write that she is obligated in one Shemoneh Esrei a day; and some write that she is only obligated in reciting a number of blessings. See Berachos 20b, and see Rashi, Ramban, Rosh, Ra’ah, who equate men and women; see also Magen Avraham 106:2; Mishnah Berurah 106:4, who write that this is the majority opinion. However, the common custom is for women to rely on the more lenient opinions, especially during when there are no shortage of children to look after and cater for. Indeed, the son of the Chafetz Chaim testified that his mother rarely prayed (during the childbearing years), and that this was the instruction issued by Chafetz Chaim himself.
The lenient opinions are based primarily on Rambam.