There's often a rogue in the family, someone who seems to love saving banner moments for wildly inappropriate times -- like the passing of a parent! When matters start to get nasty, you or your children may need to call in an arbiter, therapist and, in some cases, legal aid. It's a difficult situation, but don't despair: Very frequently, once a third party is involved, issues can be settled eventually, even if some (hopefully temporary) family fallouts are the result. It's better to call for help than to let things fester indefinitely.
When it comes to things like photographs, wedding rings and other highly sentimental heirlooms, the complications can get even more severe. This is another reason why it can be an excellent idea to talk to things over beforehand, so everybody knows what to expect and is at least somewhat prepared for the outcome, even if they aren't overly fond of it. Different families will deal with conflict in different ways, so just try to be firm yet flexible, depending on your level of interest in each item.
If you really can't settle the issue, then donation can be an option. In particular, a museum could be interested in items with historic value. It might be worth sending out feelers to see if there are any establishments willing to accept them. That solves the problem of who "wins" while at the same time allowing others to appreciate the great heirlooms left behind. If no one expresses interest, consider donating the items to people in need -- the important issue is that someone will be appreciating your heirlooms the way your family did for all those years.