Hi Ashley,
This is one of the best questions I've seen asked over here, I'm glad you asked it. It obviously isn't an easy question to answer or to ask, and I'm sorry to hear about the things that haven't gone so well in your life.
The way I would respond to that is this. Just to make this clear, this is just one person's opinion. I would make a clear distinction here:
As far as the people who knew this person go, or rather, direct family members, I would talk to them about exactly what you want to do. If you do it respectfully, I see no reason why they should be offended by a tribute to their loved one's life. I would put their feelings about your show on the highest pedestal. Your friend's passing has, unfortunately, caused everyone lots of grief as it is - this is something we cannot change now. But nothing in that friend's name should cause more grief either, no matter how well intentioned it is. Certainly, entertainment should not do so, not even a well-intentioned tribute. But, as I mentioned before, a tribute should be a celebration as much as anything else, and done tastefully, I don't see why they would be offended by the idea.
However, when we come to the general public, or people who perhaps knew the friend, or knew of them... Well. You are entertaining, and you are paying tribute in a way that expresses how you feel. Now, one of the most basic rights we have is the right to free speech. And the most important part of this right is not that we have the right to say our mind; the most important part of this right is that we have the right to say and do things that are sometimes hurtful or offensive, things that people do not like. Granted, I cannot simply abuse someone on the street - but what I can do is express an opinion that many people dislike.
What we can take from this is that no-one has a specific right not to be offended. Of course, if anyone is offended, it will probably be from a good place, be for a good reason. So at least take heart that people are good. It's good that you care, as well. But at the end of the day, the alternative is to not do what you think is right - not honour your friend, not pay tribute, not do what you want to do, and not follow your own heart, and that is too big a loss to simply take.
Good luck with your show.