The Bayit Hadash (Rav Yoel Sirkis, Poland, 1561-1640) addresses the situation of a father of a firstborn boy who gives a Kohen the money for the Pidyon Ha’ben before the thirtieth day. As the Misva of Pidyon Ha’ben sets in only on the thirtieth day, the father stipulates when he gives the money to the Kohen that the Pidyon would not take effect until the thirtieth day. The Bayit Hadash rules that such an arrangement is valid, and one may fulfill the Misva in this fashion. He adds that the father may even recite the Beracha of "Asher Kideshanu Be’misvotav Ve’sivanu Al Pidyon Ha’ben" when he gives the father the money. Even though the Misva is not actually performed yet, the father may nevertheless recite the Beracha at that point. The Bayit Hadash draws proof to this ruling from the fact that we recite a Beracha on the Hanukah candle lighting on Friday afternoon, even though we light some 30-40 minutes or so before the time of the Misva. The Misva to light Hanukah candles begins around fifteen minutes or so after sunset, but on Friday afternoon, when we are unable to light at this time, we light the Hanukah candles some twenty minutes before sundown, before we light the Shabbat candles. Nevertheless, we recite the Beracha, even though the time for the Misva has not yet arrived. By the same token, the Bayit Hadash writes, a father may recite the Beracha over the Pidyon Ha’ben when he gives the money to the Kohen before the thirtieth day.
Later writers (including the Yeshuot Yaakob) dispute this reasoning. The Sages instituted that Hanukah should be celebrated for eight days, and thus they must have taken into account the fact that at least one day of Hanukah will be Shabbat. Hence, from the outset, they established that the Friday candle lighting should be performed on Friday afternoon, since lighting candles is forbidden after sundown. As such, when we light the Hanukah candles before sundown on Friday afternoon, we are not lighting before the time of the Misva; we are lighting precisely at the time the Sages instituted that the Misva be performed. Moreover (as the Bayit Hadash himself noted in his responsum on the subject), the point of Pelag Ha’minha, according to some opinions, already marks the onset of night, as evidenced by the fact that the Arbit prayer may be recited already at that point. For this reason, too, our Hanukah candle lighting on Friday afternoon is to be viewed as being performed at the actual time of the Misva, and not before the time. Therefore, the recitation of a Beracha over Hanukah candles on Friday afternoon cannot provide proof for the situation of a father who gives money for the Pidyon Ha’ben before the time for the Misva.
This discussion yields important ramifications for the situation of Hanukah candles that are blown out before Shabbat on Friday afternoon. Generally speaking, we follow the view of "Kabeta Eno Zakuk Lah," which means that if one lit the Hanukah candles properly, with the proper amount of oil and in a location where they are capable of burning for a half-hour, he is not required to rekindle them if they are extinguished before a half-hour has passed. One might have assumed, however, that if the Hanukah candles lit on Friday afternoon are extinguished before Shabbat, they should be rekindled. Since the Misva has not yet taken effect, as it is still well before nightfall, one has not yet fulfilled the Misva and he is therefore required to rekindle the flames. In truth, however, in light of our previous discussion, this is incorrect. As we have seen, the time for lighting Hanukah candles on Friday afternoon is before sundown, and thus one who lights at that point has fulfilled the Misva. Therefore, if the candles happen to be extinguished before Shabbat, one does not have to rekindle them, just as on any other day of Hanukah. Indeed, this is the ruling of the Shulhan Aruch (Orah Haim 673:2).
Nevertheless, Hacham Ovadia Yosef writes (Hazon Ovadia – Hanukah, p. 110) that if one has time before Shabbat to rekindle the Hanukah lights, it is preferable to do so, though certainly no Beracha is recited over the second lighting.
Summary: If one lit his Hanukah candles properly and in a manner which should allow them to burn for a half-hour, he does not have to relight them if they are extinguished, even if this happens within a half-hour of lighting. This applies even on Friday, if the candles are extinguished before Shabbat, though in such a case if one has time before Shabbat he should relight them without a Beracha.