The Gemara (Shabbat 23b) explains that one who is careful regarding the Mitzvah of lighting the Menorah will be granted children who are Torah scholars. Furthermore, it is explained that Rabbi Yitzhak Sagi Nahor (son of the Raavad) inquired of Eliyahu HaNavi that many people are careful regarding the Hanukkah lights yet most people do not have scholarly children. Eliyahu HaNavi responded that it only refers to those who are meticulous regarding all the laws and intricacies of the Mitzvah of Hanukkah. Additionally, he explained that the Gemara’s blessing is four-fold: that the person himself would merit being a Torah scholar, would have children, would have male children and would have children who are Torah scholars themselves.
The Rama (Orah Haim 673:1) says that the choicest way (“Mitzvah min HaMuvhar”) to light the Menorah is with olive oil since this is essence of the miracle of Hanukkah. He adds that if one does not have olive oil, one may use any oil or even wax candles, which produce a clean and clear light. Olive oil has been used by numerous communities for generations for Hanukkah and Rabbi David Sitbon (Ale Hadas) writes that this was the custom in North Africa as well. Fortunately nowadays, olive oil is both readily available and affordable, but there is a question as to what exactly constitutes olive oil. Some products labelled as olive oil are actually the oil extracted from olive seeds, and although one would certainly fulfil one’s obligation, this would not be Mitzvah min HaMuvhar. As such, the ideal product is cold pressed extra virgin olive oil as this is most similar to what was used in the Bet HaMikdash.
Summary: Ideally one should use cold pressed extra virgin olive oil for the Hanukkah lights, but one fulfills one’s obligation with other oils and with wax candles.