Can I return for brand new iPad mini?

Assuming it was bought less than 14 days ago from an Apple Retail Store, yes you can return it without the receipt for a full refund.

If its over 14 days, then no. If it was bought form a different store, then you will have to talk to them to see what kind of return policy they have.

You will probably have to use an Apple Store or call the online Apple Store. You can always ask at Best Buy.

If you purchased it at an Apple Store the date of sale was recorded via their Point of Sale System.

You can check yourself by entering your serial number here: https://selfsolve.apple.com/agreementWarrantyDynamic.do