Question:
What happens if one remembers to make the blessings over the candles on the eighth night of Chanuka only after lighting some of the candles?

Answer:
He can still make all three blessings.  There is, however, one exception.  If the candles he already lit blew out, he does not make the first blessing, “L’hadlik ner shel Chanuka,” though he does make “She’asa nisim laavoteinu.” 

Background
The Sages instituted saying three blessings before lighting candles on the first night of Chanuka:
one over the mitzva of lighting, “L’hadlik ner shel Chanuka;”
a second over the miracle of Chanuka, “She’asa nissim laavoteinu . . ;” and
a third, “Shehechiyanu . . ,” over the joy of perfoming mitzvot that only come up periodically. 

When to say the blessings
When are they to be said?  The Maharil (Hagahot Shulchan Arukh 576) rules that all three blessings should precede the candle lighting.  However, points out Rabbi Akiva Eiger, this is only the ideal.  If however, one did not say them before, under most circumstances they can all still be said. 

She’asa Nissim and Shehechiyanu
It is clear, says Rabbi Akiva Eiger, that if one forgot that “She’asa nissim” and “Shehechiyanu” (referred to as “neis,” the miracle blessing, and “zman,” the time blessing) they can still be said. “She’asa nissim” is actually said over seeing the candles, not lighting them.  They can still be said after they are lit.  The Gemara says that one who sees candles still makes “She’asa nissim” even though he did not light them.  Similarly, “Shehechiyanu” relates to the phenomenon of experiencing a new mitzva.  That still exists after lighting.  This is also the case for other mitzvot requiring “Shehechiyanu” -- for instance, it is still said during Sukkot even if one forgot at the beginning of the holiday.

L’hadlik ner shel Chanuka
However, “L’hadlik ner shel Chanuka,” like other blessings over mitzvot, is to be said before performing the mitzva (“oveir la’asiatan”). Nevertheless, there are three reasons why the blessing can still be said in this case even though it was initially forgotten.

   1.   There are those that say that all birkot hamitzvot can even be said after the mitzva was performed (even though it should ideally be said before).
   2.   The blessing is still considered to precede the mitzva in our situation because the candles will continue to stay lit. This is the type of mitzva that is not a one-time act but an extended mitzva situation.  The mitzva lasts for the half hour or so that the candles must stay lit.  A blessing said any time before that still precedes the rest of the mitzva.
   3.   The basic level of the mitzva of Chanuka candles entails merely lighting one candle per family every night of the holiday.  The remaining candles are considered “hidur mitzva,” beautifying the mitzva.  According to the Eliahu Rabba, one even makes a blessing over the hiddur aspect of the mitzva. He speaks of a case where a family at first only lights one candle on the eighth night of Chanuka, thinking that is all they have.  If they then discover more candles and light the remaining seven, they say a blessing over this second, totally hiddur, lighting.  The Peri Chadash argues with him, ruling that if one lights six candles on the seventh night of Chanuka, mistakenly thinking it is the sixth night, and then discovers it is the seventh, one does not say a blessing over lighting the seventh candle.

Final ruling
Putting together these three approaches, Rabbi Akiva Eiger rules that when one realizes he forgot to make the blessing, he should make a blessing before lighting the remaining candles. 

However, if the candles he already lit blew out, the second component, making a blessing over the remaining time the original candles will be lit, falls off.  We are left with a dispute (Eliahu Rabba against the Peri Chadash) over whether to make a blessing.  We therefore apply the principle of refraining from making a blessing when in doubt.