The PM"G writes that ideally an onein should have his wife light ner Chanukah for him, but if that is not possible, he should light himself. Why do we not apply the usual rule that an onein is exempt from all mitzvos?
My first thought was that we see that ner Chanukah is an exception in other areas as well. Normally you don’t have to spend more than 1/5 of your money to fulfill a mitzvas aseh (ha'mevazbez al yevazbez yoseir m'chomesh), yet when it comes to ner Chanukah (O.C. 676) a person needs to even sell the shirt off his back to do the mitzvah. Just like pirsumei nisa does not allow exemptions based on an ones of money, so too, pirsumei nisa does not allow exemptions for the onein either.
Interestingly, the PMG (671:3) is not so clear when it comes to the chiyuv to spend what it takes. He writes that if there would be a monetary penalty for lighting, e.g. the state would fine you something for lighting, a person is not obligated to sacrifice more than 1/5 of his money to do the mitzvah.
The poor person who sells the shirt off his back is giving up more than 1/5 of his net worth, yet we say he is obligated to do so because of pirsumei nisa. Why, asks R’ Akiva Eiger, is paying a penalty that amounts to more than 1/5 of a person’s money any different?
R’ Baruch Mordechai Ezrahi poses the following chakirah: is the monetary cap of spending no more than 1/5 of one’s money for doing a mitzvah a din in the chiyuv of the gavra, or a din in the cheftzah shel mitzvah?
Perhaps this is the machlokes R’ Akiva Eiger and the PM"G. R’ Akiva Eiger understood that 1/5 is a cut off for the chiyuv hagavra – beyond that, a person has a right to claim ones, unless we are speaking about pirsumei nisa. PM"G, however, understood that the 1/5 cap is a shiur in spending on the cheftza shel mitzvah. If, for example, someone was charging $100 for an esrog and all you have is $500 to your name, you are at the threshold. If the same was true of the cost of Chanukah candles (pirsumei nisa), then you have no choice but to pay. It’s a different story, however, if the government decides that whoever lights has to pay a $1000 fine. That has nothing to do with the cost of the cheftza shel mitzvah – it’s simply a penalty on the gavra. In that case, says the PM"G, the rule of oneis still applies.
There may be a simpler explanation why an onein has to light. Ner Chanukah is not a personal obligation, a chiyuv hagavra, the way other mitzvos are, but is a chiyuv on the household. One of the many proofs to this idea is the fact that in hil Chanukah (67:;3) the S"A quotes a view that a child who has reached the age of chinuch may light ner Chanukah to be motzi others, but in hil megillah (689:2) the S"A writes that a katan may not be motzi anyone in megillah reading and does not quote any dissenting view. Both chiyuvim are derabbanan – why in one case does the S"A at least entertain the view that a katan can be motzi a gadol and in the other case not?
The Chacham Tzvi answers that reading megillah is a chovas ha’gavra on each individual; a katan cannot be motzi a gadol in that case. Lighting ner Chanukah, however, is not an obligation on each individual, but is rather a chiyuv on the house, "ner ish u’beiso." There is no personal obligation for the katan to be motzi anyone in – it’s just creating the state of the house having candles lit. (Compare with R’ Chaim’s chiddush discussed here that chiyuv of hadlakas menorah in the mikdash is not a chovas ha’gavra to do the lighting, but is rather a chiyuv in the chefzta shel menorah to be lit.)
Even though the onein has no personal obligation to do mitzvos, just as a katan has no obligation, he still can and should light if that is the only way his home will have the light of ner Chanukah.