Louis,
There are not many cases available at this time with a strap for iPad mini. I am an Augmentative Communication Consultant and am very familiar with your quest. Right now, your best option may be a Griffin Survivor, GumDrop Drop Tech or Otterbox Defender within a ChatBag. I have a very large Pinterest site dedicated to AAC and Assisitive Technology and Special Education with a large focus on iOS apps and accessories as well as AAC supports. You can find it here. www.pinterest.com/lasenders The options I just mentioned are on my board of iPad mini cases. The only other case with a strap thus far is the Sherpa Carry for iPad mini, but that is not super rugged.
Hope that helps.
Lauren S. Enders, MA, CCC-SLP
Augmentative Communication Consultant/Assistive Technology Consultant
Pinterest: www.pinterest.com/lasenders
Twitter: @lasenders
Facebook: https://www.facebook.com/LaurenSEndersMaCccSlp