'Activation Lock for iPhone, iPad, and iPod touch' is a security feature that's designed to prevent anyone else from using an iPhone, iPad, or iPod touch, if it's ever lost or stolen. To learn more click here --> http://support.apple.com/HT201365
For an individual to remove an activation lock absolutely requires the information for the account that applied the lock. Click on this link to learn 'How to remove Activation Lock' --> https://support.apple.com/HT201441
If you do not know the required information and you are not the original owner, get the owner of the account to remove the lock or return the device for a refund or recycle it. The device is otherwise useless to you. If the owner is unknown and has not provided contact information as a message showing in the login screen, there is no way to contact the owner unless the person who gave you the device knows them.
For help dealing with lost Apple ID or password:
• "If you forgot your Apple ID" [this is likely an email address you are using or used in the past] click here --> https://support.apple.com/HT201354
• "If you forgot your Apple ID password" click here --> https://support.apple.com/HT201487
Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Do this online at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not state what currently constitutes proof. Prior to this new online service Apple typically required the original receipt issued by Apple or an Apple Authorized Reseller and documented provenance (so typically it would be the original owner who does this, or a descendant with legal proof of ownership).
It sounds like you have tried several of the above suggestions. Help your cousin recover the information. If you cannot recover it then the answer to "What else can I do?" is recycle the device.