I have had this issue when I switched from iPhone 6sPlus to 7Plus and now the issue repeated itself when I switched from 7Plus to iPhone X.
I can’t search messages restored from backup. This same issue happened on my daughters phone.
I was able to solve it and want to share it.
1. Go to settings, then Siri & Search, then toggle off suggestions in look up and suggestions in search. Turn the phone off by holding the wake button and volume up button and then slide to off. Wait at least 2 minutes before turning the phone on.
2. After the phone is on go back to settings, then Siri & Search and turn the suggestion in look up and suggestion in search back on.
3. While still in Siri & Search scroll down to Messages. Click on it and turn Suggestion in Siri & Search to off.
4. Power down the phone and wait at least 2 minutes.
5. Turn the phone back on and open the messages app. Try to search for an old message. It will not work yet.
6. Go back to settings, then Siri & Search and scroll down to messages. Turn the Suggestions in Siri & Search for messages back on
7. Go back to messages app and try to search for old messages - it should work now.
Hope this helps some people as this has been an annoying issue for me.