If I am eating at a Chanukah party in a different place than my home, may I light the Menorah at the location of the Chanukah party and be Yotzeh?
Mishna B'rura 677:12 says someone eating at another's home but in his own city must return home to light, except if he eats there on an established basis. (Which leads to another question....)
You need to light where you live, or possibly where you're staying. Eating one meal doesn't count.
Piskei Teshuvos speaks on Mishna B'rura 677:4 about married sons who go to their parents for the night(s) of chanukah. If they intend to stay overnight, then they light at their parent's house. But if they are going home to sleep they must light at home. Then he says,
There is someone who says that if they desire to eat and drink in the company of their family (and it is not allowed to eat before candle-lighting) then they can light there even though they are going to their own home to sleep.
The source is Shu”t Kinyan Torah 5, simon 72 where he explains his reasoning.
The footnote in Piskei Teshuvos says that they must eat a proper seudah at the parent's home and just eating cake (mezonos) will not do.
Based on Dose of Halacha, it seems that according to most Poskim, you shouldn't even light with a Bracha at a party. Even according to those who say that you should light with a Bracha, it's no different from lighting in a Shul - where they light with a Bracha - and you still have to go home and light.
R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:398) writes that even though the minhag is to light the menora in Shul, one should not do so elsewhere with a beracha. Thus, if one davens in a minyan outside a Shul, or attends a wedding, there is no need to light a menora. If one chooses to do so, they must do so without saying a beracha.
R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 6:65:3) and R’ Eliezer Waldenberg (Tzitz Eliezer 15:30) explain that the mitzva of lighting the menora only applies to lighting in one’s home. Nowadays, we have a minhag to light in Shul with a beracha, too, though that wasn’t unanimous among the poskim. Thus, we cannot extend this minhag to light at parties, etc. with a beracha. R’ Shmuel Wosner (Shevet Halevi 4:65) points out that as the mitzva is to light at home, one wouldn’t fulfil one’s obligation at such a lighting. The minhag to light in Shul is reminiscent of lighting the menora in the Beis Hamikdash. Thus, even if one knows that there will be someone present who won’t be lighting at home, it doesn’t help to recite a beracha on their behalf (See Az Nidberu 6:75).
Nonetheless, others disagree. The Rivash (111) writes that the mitzva to light the menora includes lighting it at the entrance to one’s home for pirsumei nisa, to publicize the miracle. As we generally light indoors nowadays, it is important that we also participate in a public lighting. Thus, R’ Binyamin Zilber (Az Nidberu 5:37) writes that lighting in shuls nowadays ensures that we are fulfilling the mitzva properly. Lighting in a public place where others may not have lit is even more important than lighting at shul and one who does so should light the menora with a beracha.
R’ Ovadia Yosef (Yabia Omer 7 OC 57:6) writes that while most poskim write that one should light in public locations without reciting a beracha, one may do so with a beracha if they want to. Ideally, they should daven maariv with a minyan and light beforehand, as one would in Shul.
So, bottom line, you need to light again at home.