If your site has a password-protected area and its users are connected to Salesforce Contact records in your Salesforce instance, you can use the value of any field on the Contact object to prepopulate the fields on a Custom Salesforce Field included on a Petition.
To do so, insert the Salesforce Field Name as the Default Value in the following format:
For custom fields, the Field Name will always contain "__c":
Locating Field Names in Salesforce
To locate the Field Names for your Contact object in Salesforce, go to Setup > Customize > Contacts > Field Names. The Field Name is identified in the middle column of this view:
Example Use Case
If you have include the Phone field from the Lead object as a Custom Salesforce Field on a Petition, you could include the following variable as the Default Value for the field to auto-populate the phone number from the Contact record of a logged in user.