Nuala Gormley, who is 36 and from Morgan Park in Dromore, was pulled over on the Drum Road in Cookstown on Tuesday evening.
Police said components of an improvised explosive were seized from the vehicle which was at the centre of an overnight security alert.
Detectives told Dungannon Magistrates' Court on Friday that firing mechanisms and remote signalling equipment were discovered in an empty oil container inside the car.
They said Gormley has admitted her support of republicanism.
A detective constable said: "Provisional examination by Forensic Science Northern Ireland, indications suggest that the relevant items seized are the component parts of an improvised explosive device. She did have gloves, wellington boots and two coats."
Gormley was refused bail and will reappear at Dungannon by video-link on 6 September.