Mazel tov on your son's birth!
Most breastfeeding women prefer to first try progestin only pills as they have less of an effect on the milk supply. By taking the pill at the same time each day, which protects against a dip in progestin level, a woman can help reduce the chance of stains. Estrogen-containing pills can decrease a woman's milk supply, even if it is well-established. However, the decrease does not happen to all women, so they remain an available option if the staining does become a problem.
Please remember that it can take about two months for the body to adjust to any hormone treatment. You might want to consider starting the pill after you first immerse in the mikveh to avoid the need to do bedikot while adjusting to the pill. However, you will not have the contraceptive benefits of the pill until after about ten days of pill use. Your chances of getting pregnant within six months of childbirth, if you have not yet resumed a period and are exclusively breastfeeding are about 1% (see article on the site about the lactational amenorrhea method). You may wish to add a spermicide, to reduce these chances even further.
There are no specific leniencies for stains on the pill. We highly recommend reviewing our article on ketamim to learn what leniencies apply to stains in general.
Another contraceptive option, which neither affects the milk supply nor causes staining, is the diaphragm. This method is more halachically controversial. Rav Yehuda Henkin, the halachic supervisor of this website, permits the diaphragm and considers it halachically preferable to methods that are likely to lead to prolonged staining, such as the progestin only pill.