I'm sorry you are in such as difficult situation, especially as you must still be grieving.
As you own the house and they will be paying you rent, it seems to me that some sort of formal agreement is needed to protect both sides.
I suggest you get advice from a solicitor (free initial consultation, maybe?) or Citizens Advice. If you do sign over their share, make sure that you are protected. If I remember rightly, when we shared a house with my parents we were advised to own it as 'tenants in common' - that way neither side could sell up and leave the other homeless.
Making sure both you and they are protected shouldn't be controversial, unless they're aiming to do something underhand. I hope you can get the right advice and organise things so that you are all happy.