My daughter was prescribed Yasmin for her heavy menses, however we have a strong family history of?

Latest studies suggest there is no link between oral contraceptive use and increased risk of breast cancer but there has been conflicting evidence along the way...
from http://www.drugs.com/sfx/yasmin-side-effects.html

Oncologic side effects
"...The World Health Organization committee also noted that some studies "have found a weak association between long-term use of oral contraceptives and breast cancer diagnosed before the age of 36, and perhaps up to the age 45....It is unclear whether this observed association is attributable to bias, the development of new cases of cancer, or accelerated growth of existing cancers."

The World Health Organization committee further concluded that there is no increased risk of breast cancer in women over the age of 45 who have previously taken oral contraceptives. In addition, studies suggest that use of oral contraceptives does not place specific groups of women (like those with a family history of breast cancer) at higher or lower risk, and variations in the hormonal content of oral contraceptives do not influence the risk of breast cancer..."

There is also some great information at the
Oral Contraceptives and Cancer Risk: Questions and Answers page at
http://www.cancer.gov/cancertopics/factsheet/Risk/oral-contraceptives

How do oral contraceptives affect breast cancer risk?

A woman’s risk of developing breast cancer depends on several factors, some of which are related to her natural hormones. Hormonal factors that increase the risk of breast cancer include conditions that may allow high levels of hormones to persist for long periods of time, such as beginning menstruation at an early age (before age 12), experiencing menopause at a late age (after age 55), having a first child after age 30, and not having children at all.

A 1996 analysis of worldwide epidemiologic data conducted by the Collaborative Group on Hormonal Factors in Breast Cancer found that women who were current or recent users of birth control pills had a slightly elevated risk of developing breast cancer. The risk was highest for women who started using OCs as teenagers. However, 10 or more years after women stopped using OCs, their risk of developing breast cancer returned to the same level as if they had never used birth control pills, regardless of family history of breast cancer, reproductive history, geographic area of residence, ethnic background, differences in study design, dose and type of hormone, or duration of use. In addition, breast cancers diagnosed in women after 10 or more years of not using OCs were less advanced than breast cancers diagnosed in women who had never used OCs. To conduct this analysis, the researchers examined the results of 54 studies. The analysis involved 53,297 women with breast cancer and 100,239 women without breast cancer. More than 200 researchers participated in this combined analysis of their original studies, which represented about 90 percent of the epidemiological studies throughout the world that had investigated the possible relationship between OCs and breast cancer (2).

The findings of the Women’s Contraceptive and Reproductive Experiences (Women’s CARE) study were in contrast to those described above. The Women’s CARE study examined the use of OCs as a risk factor for breast cancer in women ages 35 to 64. Researchers interviewed 4,575 women who were diagnosed with breast cancer between 1994 and 1998, and 4,682 women who did not have breast cancer. Investigators collected detailed information about the participants’ use of OCs, reproductive history, health, and family history. The results, which were published in 2002, indicated that current or former use of OCs did not significantly increase the risk of breast cancer. The findings were similar for white and black women. Factors such as longer periods of use, higher doses of estrogen, initiation of OC use before age 20, and OC use by women with a family history of breast cancer were not associated with an increased risk of the disease (3).

In a National Cancer Institute (NCI)-sponsored study published in 2003, researchers examined risk factors for breast cancer among women ages 20 to 34 compared with women ages 35 to 54. Women diagnosed with breast cancer were asked whether they had used OCs for more than 6 months before diagnosis and, if so, whether the most recent use had been within 5 years, 5 to 10 years, or more than 10 years. The results indicated that the risk was highest for women who used OCs within 5 years prior to diagnosis, particularly in the younger group (4).

References:
(2) Collaborative Group on Hormonal Factors in Breast Cancer. Breast cancer and hormonal contraceptives: Collaborative reanalysis of individual data on 53,297 women with breast cancer and 100,239 women without breast cancer from 54 epidemiological studies. Lancet 1996; 347:1713–1727.
(3)Marchbanks PA, McDonald JA, Wilson HG, et al. Oral contraceptives and the risk of breast cancer. New England Journal of Medicine 2002; 346(26):2025–2032.
(4)Althuis MD, Brogan DD, Coates RJ, et al. Breast cancers among very young premenopausal women (United States). Cancer Causes and Control 2003; 14(2): 151–160.

Hope this helps. Good luck.