The iPad 2, 3 and 1st generation iPad Mini are all ineligible and excluded from upgrading to iOS 10.
They all share similar hardware architectures and a less powerful 1.0 Ghz CPU that Apple has deemed insufficiently powerful enough to even run the basic, barebones features of iOS 10.
Why would you want to upgrade, anyways?
Since iOS 8, older iPad models such as the iPad 2, 3, 4 and the original iPad Mini have only been getting the most basic of iOS features.
Older iPad users get none of the newest and coolest cutting edge features that newer model iDevices, with better hardware specs, can handle with more aplomb.
FYI,
Just because your 1st gen iPad Mini is no longer able to upgrade to the latest iOS doesn't mean your 1st gen iPad Mini is suddenly useless.
So, your options are really keep using your 1st gen iPad Mini or purchase a new iPad if you feel you need or want to.
Your current 1st gen iPad Mini will still work and continue to receive app updates until sometime this Fall 2017, when, with the introduction of iOS 11, ALL support for older 32 bit iDevices and any iOS 32 bit apps will come to a final end.
Your 1st gen iPad Mini will still work and function as it always has, but will no longer receive any more app updates after some time in the Fall of 2017.
The 1st gen iPad Mini has had 4-5 years of iOS upgrades/updates
The final app updates your 1st gen iPad Mini will receive will be its last!
Currently, your 1st gen iPad Mini is STILL receiving app updates. Look for this to change sometime in the near future
This is why Apple introduced the new, lower cost 2017 iPad 5 model, in April
Apple is trying to get older iPad 2, 3, 4 and 1st gen Mini iPad users to upgrade to more modern iPad hardware in anticipation of iOS 11, which will render all older 32 bit iDevices obsolete.
Good Luck to You!
