How can I add wife's Gmail account to her new iPad mini?

If you want to use the Mail app to send/receive gmail, see the "Add an email account" section of Add and remove email accounts on iPad - Apple Support

Another option is to go the App Store on the iPad and download the gMail app and use it instead of Apple's Mail app.

Richard Thomas wrote:

is there a phone icon for the iPad?

No.

Do you have an iPhone, or just an iPad?

An iPad, whether a Cellular model or not, is not capable of directly accessing Cellular Voice or Messaging features. An iPad with Cellular connectivity alone can only access IP-data services. An iPad is not, and can never be, a standalone substitute for an iPhone.

However, when coupled with an iPhone, the capabilities of iPad significantly broaden to include access to Cellular Voice and Messaging services of the associated iPhone - relayed to the iPad over WiFi using Apple’s Continuity features. For the most part, an iPad using Continuity is able to access Cellular calling features, of the iPhone, when both the iPad an iPhone are in close physical proximity and are connected to the same WiFi network.

If your both your Cellular Carrier and your calling plan support WiFi Calling, your iPad may be able to make and receive calls independently of the iPhone - even when the iPhone is switched off, elsewhere, or using a different WiFi network. Not all carriers support full WiFi calling.

The settings that you need to change/verify on your iPhone and iPad are:

On your iPhone

Settings > Mobile Data >WiFi Calling > 

• WiFi Calling on This iPhone - set to ON
• Add WiFi Calling for Other Devices - set to ON (if available)

Settings > Mobile Data > Calls on Other Devices > 

• Allow Calls on Other Devices - set to ON
• [Allow Calls On] - explicitly enable your new iPad from the list of devices.

On your iPad

Settings > FaceTime > Calls from iPhone > Calls from iPhone - set to ON

Additional information about Continuity and the relevant Cellular services can be found here:

Continuity

Work across all of your devices seamlessly – Apple Support

SMS/MMS Messaging

Forward SMS/MMS text messages from your iPhone to your Mac or iPad - Apple Support

WiFi Calling

Make a call with Wi-Fi Calling – Apple Support

Making and receiving calls on iPad

Make and receive phone calls on iPad - Apple Support

When correctly configured, the iPad FaceTime App provides access to Cellular Voice Calling features - whilst Messages supports SMS/MMS messaging services.

Can you give a few more details? Are you trying to access it through the Apple Mail app? Or a different app or browser? Are you sure you are using the correct username and password?