The gemara (Menachos 89) derives from the words "mei'erev ad boker" that menorah needs to have 1/2 a lug of oil per ner so that it can burn the entire night.
Is this a tnai in the chiyuv hadlakah, that unless there is enough oil one has not fulfilled the mitzvah to light the menorah, or is it a separate chiyuv, i.e. there is a chiyuv hadlakah to light the menorah, which we learn from "l'ha'alos ner tamid," and on top of that there is a new chiyuv of "mei'erev ad boker" to see to it that the menorah burns throughout the night?
Nafka mina: what if the menorah was originally lit with less oil, but the kohen added more oil later so the ner burned through the night?
If the shiur of 1/2 lug is a din in hadlakah, it's not a good hadlakah. If it's a new din, then the hadlakah was good, and "mei'erev ad boker" is fulfilled so long as the ner burned until morning.
Tos Rosh (Shabbos 21) writes that when the Chashmonaim found that one jug of pure oil, they divided it into 8 portions to last 8 nights. He then asks how they could do this -- don't you need 1/2 a lug to light? Tos Rosh answers that that requirement is mandated only when possible, but is not m'akeiv b'dieved when there is no other way to light.
R' Noson Gestetner explains that the Tos Rosh holds like the second tzad of our chakirah, that there are two dinim, a din hadlakah and a separate din of mei'erev ad boker. By dividing the oil into 8 parts there could be at least a kiyum of hadlakah on each night, even if that second kiyum of burning the entire night would not have been fulfilled if not for a miracle.