This can be a tough one to answer on a general level, since the answer can be different depending on where the cysts are, how big they are/how long a person has had it, whether it is related to hormones, and the person in general (what the overall health picture is).
However some safe herbs that may help are: Calendula, Cleavers, & Marshmallow root (however this is more of a topical one). Herbs that support lymphatic movement and drainage are a good place to start. Another group of herbs that can help are circulatory support herbs (Turmeric, Cayenne, Cinnamon, Prickly Ash- Zanthoxylem spp., etc.).
Some other aspects to consider: Full hydration and limiting sugar, wheat, and dairy in the diet.
As mentioned above, there are other herbs as well but based on the person and what type of cyst it is. These are some general herbs and approaches, but I hope they is still of some support.