How to reset my iPad mini when it is signed into a different Apple ID?

Without the Apple ID and Password of the original user, you will not be able to reset or use the iPad. You will encounter Activation Lock which requires the info of the original user. Check this link...

How to remove Activation Lock - Apple Support

Presumably you are being prompted for your AppleID credentials - so as to satisfy an Activation Lock that is intended to prevent reuse of the device by anyone other than the legitimate owner. 

To do so you will require the AppleID an associated password that was last used to successfully activate the device.

If you have acquired a previously used iPad, if the required AppleID credentials are not your own, then your will require the assistance of the former owner - as without their assistance the iPad is (and will remain) useless to you.

Here are the Apple support pages that outline the Activation Lock - and the only mechanisms by which it can be satisfied:

Activation Lock for iPhone, iPad, and iPod touch - Apple Support

Turn off Activation Lock – Apple Support

As you will discover, unless you are the owner of the AppleID that was last used to activate the device, you’ll not be able to defeat or bypass the Activation Lock. With documentary evidence that you are the original owner (such as sales receipt showing the serial number), Apple may be prepared to assist - otherwise they and nobody here will be able to help you.

If you have the necessary documentation, Apple have recently introduced a portal though which you may be able to request release of an Activation Lock:

How to remove Activation Lock - Apple Support & start an Activation Lock support request.

However, recovery of the AppleID credentials should be possible…

If you have forgotten your AppleID or associated password, these can be recovered here:

https://iforgot.apple.com/

To recover your credentials, you’ll need access to any one of the following:

• Your primary email address mailbox that corresponds with your AppleID
• Any of the secondary/recovery email addresses that should be configured for your AppleID account
• Any of the trusted telephone numbers (fixed line or Cellular/Mobile) that are associated with your AppleID

Unless you AppleID account has not been fully/correctly configured, or has been seriously neglected, recovery of the AppleID and password should not be difficult.

More information about recovery of your AppleID:

If you forgot your Apple ID - Apple Support

More information about recovery of your AppleID password:

If you forgot your Apple ID password - Apple Support

When you recover access to your credentials, you would be well advised to log-in to your AppleID account from a web browser - and verify/update any email addresses and trusted telephone numbers:

https://appleid.apple.com/

olson85 Said:

"iPad mini: how to reset my iPad mini when it's signed in a different Apple ID."

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Where exactly did you get this device? Has it always been you device, or is this a used device that you purchased?

Troubleshooting a Locked iPad:

Have the Seller Log Out of This:

Learn from this:

If this is a used device, then the person currently signed in to the iPhone needs to log themselves out of it. Elsewise, it is useless to your account.

This is the site the seller should have used, prior to selling it to you: What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support. The seller should have known better.

So, return it, requesting that they log out of it, and use that post I just provided prior to handing it back in.