How to get ongoing notifications of unread texts?

The closest you’re going to get is to set Messages alerts to repeat 10 times by going into Settings → Notifications → Messages. There, swipe to the bottom to reveal the Repeat Alerts menu:

This will cause Messages notifications to ring for 20 minutes after the initial one, or until you’ve read/dismissed them.

Next, make sure Do Not Disturb is disabled by going into Settings → Do Not Disturb and toggling both Manual and Scheduled to Off (there are no VIP notifications in Messages yet).

You could also assign a custom loud and obnoxious text tone to the family member in question that will make it more likely a message from them will wake you up, but that alert sound will trigger for all text messages they send you, even during the daytime.

Coworkers who were on-call 24/7 also found it useful to turn on LED Flash for Alerts under the HEARING section of Settings → General → Accessibility.

If you lay your iPhone face down at night it will flash brightly when you receive a notification, although this will be active at all times for all notifications, so you might need to disable this every morning and re-enable it at night.

A note on Do Not Disturb: if you have a habit of using Do Not Disturb and occasionally forget to turn it back off, you can make sure it gets disabled at night by enabling Scheduled for a 1-minute duration every evening, say, from 10:30 to 10:31 p.m., or sometime before you go to bed.