Inside your form, you want to display existing cases in a checklist or single select question. To do this, you need to turn enable an Add On called "Custom Single and Multiple Answer Questions". This allows you to display custom choice lists in forms (i.e. a list of case sharing groups) instead of just items from a Lookup Table.
Go to your Application Settings, then Select the Add-Ons tab.
Choose the button with three lines on the right-hand side that is next to the save button and select 'Calculatons'
Turn on the "Custom Single and Multiple Answer Questions" add-on to enable the necessary feature:
Step 2: Configure Your Form
In your form, add a new Checkbox Lookup Table question
Choose the Lookup Table Data item below the added question to configure the data that will be displayed in the question. Choose the (...) button and enter the following:
Set the query expression to instance('casedb')/casedb/case[@case_type='your_type'] (Replacing your_type with the case type you want to display)
Set instance ID to casedb
Set the Instance URI to jr://instance/casedb
Hit the Save button
Set the value to @case_id
Set the label to case_name
If you would like to restrict the cases shown to be only child cases of the chosen case which you are updating you can add the following to the end of the query: [index/parent = instance('commcaresession')/session/data/case_id]
If you would like to restrict the cases shown to be only open cases you can add the following to the end of the query: [@status='open']