Apple - Can iOS display a text message notification icon in the status bar?

Notification icons in the status bar aren't a feature of iOS, just Android, so there's no way to get the look you're talking about.

You will, however, get a notification and a sound on the lock screen/home screen when you get a text, a badge on the Messages app icon that you can enable/disable in Settings -> Notifications -> Messages, and the notification will be accessible in Notification Center (which shows up when you swipe down from the top of the screen) until you read the message.

While iOS surely has some advantages over Android, its customisation options aren't one of them: there's just no way to get those icons you want there.

Though I haven't tested it, it may be possible to do this if you jailbreak an iPhone—see @timothymh's answer.

tl;dr: There's no way to do this with iOS.

Unfortunately this is not possible as of iOS 10.3.

By default iOS has this behavior when it comes to message notifications:

• new messages will be displayed on the Lock Screen (called Notifications)
• the notifications on the Lock Screen disappear if you unlock the phone
• the number of unread messages will be displayed as a badge on the Messages.app's icon
• you can see all notifications for unread messages by pulling the Notification Center (swipe down from the top of the screen)
• if you read a message by opening the corresponding app the notification will disappear from the Notification Center

I hope this helps.

If your phone is jailbroken, you can install a tweak which modifies banner notifications such that they appear in the status bar, such as ThinBar or Resero. There are also a plethora of tweaks which let you display icons in the status bar when an app icon has a badge on the home screen.

Otherwise, this is not possible.

Edit: As Monomeeth mentioned below, jailbreaking is a somewhat technical and sometimes buggy process, which in many cases voids the warranty, so I would not recommend jailbreaking the phone of someone who isn't utterly familiar with what that means, such as (I assume) your mother.