In any checklist, you can set the Custom Rarity for a bird by tapping on the bird name to go into the detail view, then tapping where it says "Rarity: " in the lower right, as shown here:
Once a list contains any birds with Custom Rarity, it can then be sorted by Custom Rarity via the Checklist Sort Style menu:
Birds without Custom Rarity set will show up at the end of the list under "Uncategorized." Any checklist templates you make based on your list will inherit the Custom Rarities you set.
If you would like to create your own checklist templates containing Custom Rarity to share with others, all you need to do is create a .csv file with three fields: Common Name, isABird, and Rarity. For example, here's a minimalist list containing rarities:
Mallard,1,1 Black Phoebe,1,2 Zone-tailed Hawk,1,6
Creating a file with those entries and importing into Birdathon would give you a checklist with three species with Custom Rarities of 1, 2 and 6 respectively. The "isABird" field is used in case you want to make a checklist template with non-bird items (such as butterflies or mammals); non-birds are 0 and birds are 1.