Changed phone number to new contact but messages still shows old contact
Months later, with no better answers in sight, I decided to try deleting my own contact card (where "Daughter"'s number had been used as mine at one point but had since been removed).  I'd been trying to be more surgical about the problem and wanted to avoid this big hammer.  It solved the problem.  

The detailed steps I took are below.  Some of the steps are probably unnecessary but I really didn't want to take a chance.  And you could probably do it fine without needing MacOS Contacts connected through iCloud but I found it made it easier to reimport and know I got exactly my contact info back the way it was (and I don't know how I would have done that just within iOS).  


Started with iMessage off on the phone but iCloud active for
Contacts on both phone and MacOS
In linked Contacts on MacOS, exported my own card - the one with the
problem - to the Desktop as a vcf file
On the phone, deleted my own contact; verified it went away from my
MacOS contacts through iCloud sync
On the phone sent message to Daughter; confirmed it stayed saying
"Daughter"
On MacOS, imported my vcf file into Contacts; confirmed the card
reappeared on the phone through iCloud sync
On the phone sent message to Daughter; confirmed it stayed saying
"Daughter"
On MacOS, used the Card -> Make This My Card; confirmed that the
silhouette showed up on my card on the phone through iCloud sync (discovered you have to not have your own contact open on the phone for this to sync)
On the phone sent message to Daughter; confirmed it stayed saying
"Daughter"
Turned on iMessage on the phone and accepted connection on MacOS
 for the phone number
On the phone sent message to Daughter; confirmed it stayed saying
 "Daughter"