Tattoo for mental health purposes

Contemporary halachic authorities have recognized some psychological illnesses with potential threat to life as a form of pikuach nefesh. (See Igros Moshe E.H. 1:65 regarding severe post-partum depression; Tzitz Eliezer 4:13(3) regarding undergoing a dangerous surgery to heal a mental illness.)
In the classic case of pikuach nefesh, such as eating on Yom Kippur, if the patient is definitive about the need to eat, we accept his opinion even if the doctors do not think it necessary. (Shulchan Aruch O.C. 618:1)
Thus, if there is real concern that the patient will endanger herself if she is not able to receive the "treatment" she considers necessary – there is basis to permit violating the prohibition of tattooing to save her from potential self-harm.
However, it is questionable whether the reliance on a patient's determination to eat on Yom Kippur can be extended to other forms of pikuach nefesh, especially to a mental patient who is not in a proper frame of mind. Therefore,  this should only be done in consultation with a religious psychologist or psychiatrist who can evaluate that the concern for life is real and that getting the tattoo will calm her. You can try contacting Nefesh Organiztion, The International Network of Orthodox Mental Health Professionals. [In Israel, you can contact Dr. Avigdor Bonchek.]
You should also explore other possibilities, such as using micro-pigmentation (semi-permanent makeup) rather than a classic tattoo. According to many authorities this prohibition is rabbinic, and can more easily be allowed in this case. (See the article: The Tattoo Taboo and Permanent Makeup Too, which might be helpful.)  

Best Wishes,

Meir Orlian