You are going to probably get different answers from each attorney on this. There is no one way. Here are a couple of examples.
You should separate the items of monetary value from the items of sentimental and little value. The monetary items should be appraised but not required. If there are a lot of items you may also want to arrange them in agreed upon groups.
As for selection, you can put names or numbers in a hat. The names or numbers are chosen randomly to determine the order of picking items. You would then each pick an item in a round.
In round two, you can do one of two things - reverse the order so the person picking last in round one now picks first, or keep the same order. I prefer reversing the order.
Good luck. The more everyone gets along, the easier this will be.
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Some other solutions clients have included rolling dice to pick order. We have also had a third party come in and lump into piles and then draw for the piles. An antique dealer works best for that to get some parity on values.
this is not legal advice for your specific situation. Contact a lawyer of your choice to establish an attorney client relationship to address your specific needs.
I would suggest that each beneficiary prepare privately a list in order of priority of what they want first, second, third, twentieth, etc. Have each person do this and then have the group sit down after obtaining the list of all items. If the number 1 choice is everybody then that is what needs to be appraised. Then go to the next item and see if but one erson wants it and no others. This would be a process of elimination where the sentiment vs. Value oriented desires become clear. If the list can be narrowed down form 20 items to 5 then the final task will be easier.....
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained.