It is very common for young married couples to visit the parents or in-laws during Hanukah, and to stay there overnight or for several days. In such a case, the couple should not light Hanukah candles, and they instead rely on the lighting of the parents. Their obligation is fulfilled through the parents’ lighting, and thus they do not have to light, and they most certainly should not recite Berachot over lighting.
However, if a couple spends Shabbat with parents, and they plan on returning home after Shabbat, then on Mosa’eh Shabbat they should light Hanukah candles when they return home. Even though they are still in the parents’ home when the parents light Hanukah candles right at the conclusion of Shabbat, they do not fulfill their obligation with that lighting, since they plan on returning home that night. They must therefore light Hanukah candles – with the Berachot – when they arrive home.
An interesting Halachic question arose many years ago in Europe, when it was common for people to travel long distances by train. If a person rode during Hanukah on overnight trip, and would not reach his destination until the next day, would he be required to light Hanukah candles on the train? The Halachic authorities ruled that indeed, for such a person, his seat or cabin in the train is considered his "home" with respect to the obligation of Hanukah candles. There is no source indicating that the Misva applies only to a stationary home; even in a "home" that moves, one is obligated to light Hanukah candles. Thus, when a person purchases a ticket for the train, effectively "renting" his seat or cabin, and the train travels through the night, his seat or cabin is considered his home for that night and he must light Hanukah candles on the train. (Of course, this assumes that this is possible and legally acceptable.) Interestingly enough, it is told that in Czechoslovakia, where apparently many Jews traveled by train, when an overnight train passed by during Hanukah, it appeared like a beam of light, as all the Jews on board lit Hanukah candles by their windows.
Of course, this Halacha is not so applicable nowadays, when people do not commonly take overnight train rides, but it is nevertheless interesting to note that in principle, the obligation of Hanukah candles applies even during travel. (See Yalkut Yosef – Hanukah, p. 169.)
Summary: If a couple spends the night in their parents’ home, they do not light Hanukah candles. If they spend Shabbat with parents and plan on returning on Mosa’eh Shabbat, then on Mosa’eh Shabbat they light Hanukah candles when they return home. Somebody who takes an overnight train ride during Hanukah would, theoretically, be required to light Hanukah candles on the train, if this was possible and legally acceptable.
