Can iOS display a text message notification icon in the status bar? My mother has just moved from an Android phone to an iPhone SE and she misses the fact that she no longer gets a text message notification icon at the top of the screen. This is a small envelope shaped icon in the same region of the screen as the WiFi connected symbol.

Is there any setting or application that will give her this functionality?
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.