Are there any references for the left over drops of oil left in the menorah cups to be used for refuah?
There is a reference regarding using the leftover oil to smear on to a person that has cellulitis (a “rose”). It is brought from the Kotzker Rebbe that he said that there is a hint to this in the term, “uminosar kankanim naaseh nes lashoshanim”, lit. from the left over of the oil there was a miracle that happened to the Jews, (referred to as roses, as it says in Shir Hashirim, that the Jews are compared to a Rose). But if the word shoshana is translated literally it would hint to a “rose”.. in someone’s foot.
Others take issue with doing such a segulah, because we are not allowed to benefit from the leftover oil. Others say that it would be permitted to use the leftover oil from the lights that already burned a half hour, (which can be considered according to some as if it isn’t leftover oil of the actual mitzva), but not from leftover oil from the first half hour. Everyone agrees that one may use the oil leftover in the bottle for refuah if one wants to.
Other than that I haven’t seen that it is a segulah for other ailments
Ozer Yisroel pg.116 in the name of R’ Y. Zilberstein shlit”a, Dinei Hadlokas Neiros Chanukah R Dunner 10-6.