If you visit Hanuman temple especially on Saturdays, we see a number of devotees offer oil (especially gingili oil). Why oil is offered to Lord Hanuman
Black sesame or Til seeds are one of the materials signified by Saneeshwara ( Lord Saturn). Gingelly oil derived from sesame seeds are therefore used to burn oil lamps on Saturday to propitiate Saneeshwara, especially if the devotee is having sade-sati.This is a wide practice in southern India, called Neeranjanam lamp for Saneeshwara. Due to reasons given in this answer, Lord Hanuman alleviates the trouble of devotees having bad effects of Saneeshwara / sade-sati. This would be basis of offering gingelly oil to Hanuman on Saturday. Again these are local practices based on local fables ( desa kala pramanas) and may probably have no scriptural basis.