Dress nicely for the funeral and send a card. The widow will likely have a lot of help for the first few weeks or so. I would think long term. Maybe after the hustle and bustle of the funeral has died down, you could invite her over for lunch or tea or an afternoon visit. Or go to her house. Be willing to listen to her share stories about her late husband. I think the loneliest time after a death is 3-9 months. The initial impact is gone and everyone remembers the anniversary, but the months in-between get lost.
After my step mom died (I was with dad for the first week or so afterwards), I made sure I called twice a week for the next several years. There were days, weeks even, that it wasn't much more than, "Hi, Dad! How's it going? Love you! Bye!" but he swears that, at times, that's all that kept him connected. Perhaps you could be that person for your neighbor.
But, ultimately, follow your heart. Random acts of kindness really work.