Start with the death certificate. Get a copy from the state or county where he died. That might very well give you his mother, father and their birth places.
Then look for an obituary. If there is one, it should be in the newspaper where he lived at the time of death. It's also possible that his home town newspaper published an obit, if he was from somewhere else. An obituary will usually give you siblings (and sometimes, nephews and nieces) names and area where they lived.
If it's a city, then go to (or call) and look at their old phone books for those names. Lots of people still live where they did 35 years ago, so you might get lucky. You can try current phone books for the names. Then you can try facebook, etc. for those names.
Have you considered asking your late husband's mother for information? Or maybe her siblings? Families usually know something, if you ask.