Indeed, the Chanuka lights must last quite a while on Friday evening. There is a disagreement regarding whether the mitzva to light usually begins at sheki’ah[1] or at tzeit hakochavim.[2] [3] Chanuka lights should remain lit half an hour past that point.[4] Since we generally accept Shabbat around 20 minutes before sheki’ah and Chanuka lights must be lit prior to that time, the lights must last considerably longer than on a weekday.
Generally, if one set up the Chanuka lights with enough fuel to last the required time but unexpectedly they went out earlier, he does not have to relight them.[5] The question is whether this applies even when lighting them for Shabbat, which has to be done during the day and sometimes, as this year, before Chanuka has even begun. If the lights go out before Shabbat begins, it might be worse than the situation on an ordinary night, when at least the mitzva of lighting is fulfilled at the correct time.
The Terumat HaDeshen[6] makes an interesting argument on the matter. Although the purpose of the lighting on Friday is for the lights to burn into the night, the lighting itself is an important part of the mitzva, specifically the hechsher mitzva (the necessary facilitator). He notes that we recite the beracha when we light, although it is still not the mitzva’s classic time, because it is necessary to carry out the lighting prior to the approach of Shabbat. Thus, however long the lights end up burning, including during the period before Shabbat, is sufficient.
According to the Terumat HaDeshen, even if it is possible to relight the extinguished candlles before Shabbat begins, it is not necessary to do so. (We note that several poskim say that although it is not necessary, it is still worthwhile to relight the lights. This is true even during the week, when the the lighting was performed at the correct time, and certainly before Shabbat.[7]) Certainly, then, you should not feel that you have failed halachically if the lights accidentally go out sometime after Shabbat has begun, when you cannot relight them.
The Taz (673:9) takes issue with the Terumat HaDeshen and requires relighting an extinguished Chanuka candle if one has not yet accepted Shabbat. He agrees with the Terumat HaDeshen’s primary thesis – that the mitzva can be actuated before nightfall. However, he says that the fulfillment of the mitzva begins at the last moment that one can light the candles, which is immediately before Shabbat starts, either automatically at the conclusion of Friday or at whatever time one accepts Shabbat earlier. If the candles are still burning at that point, it is equivalent to their burning a little into a regular night of Chanuka, when it would not be necessary to rekindle them. However, according to the Taz, one cannot be credited with the mitzva before Shabbat has actually begun for him. Thus, if the Chanuka candle is extinguished before Shabbat begins, one must light it again. In the case with which you are concerned, in which the flame goes out after your Shabbat has begun, the Taz would agree that you would have fulfilled the mitzva already.
To minimize your nervousness that your Chanuka lights might not last as long as intended, we suggest you consider the following. First of all, after the first day of Chanuka, it is pretty safe to assume that at least one light will last long enough, and that is sufficient according to the basic halacha,[8] which requires just one light per household.[9] Even on the first night, if more than one adult is lighting, you can instruct everyone to have in mind that if some lights go out prematurely, the remaining one(s) should count as a household Chanuka light. Again, in no way is this required halachically, but it may assuage your worry.
In particular, a practical idea is to use a (long-lasting) wax candle rather than olive oil for this night,[10] as wax is usually more reliable. Another piece of general advice is to do a trial run to see that your wick/oil combinations work well. However, just as importantly, we suggest that you get used to following the normal halachic rules without being more nervous than halacha expects or the Torah desires.