The term Ariadne's thread used in science is derived from the incident in Greek mythology described in answer above. As per this legend, Theseus left a trail of his passage through the complicated and confusing network of passageways of the Labyrinth, using thread from a ball of thread given to him by Ariadne.
In science this terms refers to method of solving any problem by keeping a trail of all the options available and tried out at all the times. This trail in Science is, usually in the form of records. However, other physical means of keeping the trail can be used also. This method of solving problems is generally contrasted with the method of "trial and error".
Over time, the phrase "Ariadne's thread" has received various connotations, being till today, an essential mythological and cultural point of reference, giving rise to many interpretations.
Legend says that in Greek mythology, Ariadne is the daughter of King Minos and Pasiphae queen.
When Theseus arrived in Crete to fight with the Minotaur, Ariadne, who had fallen in love with the hero, helped him to get out from the famous maze of twisting corridors built by Dedal. She gave him a piece of thread to show him the way to return. As to escape her by the wrath of her father, Minos, Theseus promised Ariadne to take her to Athens. But he left her on the road, on the island of Naxos, where, as a variant claimed, was killed by Artemis.
According to another legend, Dionysus, who was passing there, would be enamored of the young girl and would have taken her of marriage. As a wedding gift, he offered her a golden crown, forged by Hephaestus - considered to be the god of fire, crown which was later transformed into a constellation.
It also issued the hypothesis that there were two women with the same name: one who helped Theseus and another who was married to Dionysus. In any case, Ariadne - primitive fertility goddess - was worshiped in Athens, Naxos and Cyprus.