The simple answer to your question about suing is, yes,, you can. Possibly you could get a restraining order against the neighbor or maybe even an injunction against the landlord. Whether you want to do so is an entirely different question however. I think the Court would be disinclined to find with you, especially if the neighbor denies smoking. And these remedies could be expensive, and certainly they will be very distracting and, at best, unsatisfactory in the long run.
You have other remedies, however. You may withhold in an undetermined amount of rent when a landlord fails to remedy a material habitability issue. Almost certainly this will precipitate an eviction action against you, though. It would be in that eviction suit that the habitability defense would come into play. Again, though, you have a problem of proof here. Apparently the landlord believes the neighbor and would prefer that you vacate instead of the neighbor. I, as an impartial observer, cannot discount the landlord's position out of hand; neither will the judge. You better have strong evidence not only that the neighbor is, indeed, smoking but also that the smoke causes you and your family material distress. Also you will have an eviction on your record, something that could haunt you in the future even if you win.
The best suggestion I have is for you to look for a more suitable place to live. Yes, you have remedies, but none of them is as practical as simply moving to a place where you can live in quiet enjoyment of your home. Distasteful, expensive, and distracting as this may be, probably it still is your best remedy.
Yes, you can sue, but your chances of winning are slim if your legal basis is breach of quiet enjoyment. Your remedy is probably to move out and live elsewhere. It is not likely the court will award you any monetary damages.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This Avvo.com posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, please consult with your own attorney.