Your OUPath should be comma separated, not semi-colon separated. But that bit worked and the computer landed in the right OU? If so, odd but go with it :)

I wouldn't be surprised if you had to reboot before the changed name was relevant as far as "Add-Computer" is concerned.

This, however, assumes that Rename-computer is running without throwing red error messages everywhere. Is that a fair assumption?

Try to do it all in one go with the Add-Computer and its "NewName" argument.
And you'll need to use commas instead of semicolons for the OU path.

$serial = Get-WmiObject win32_bios | select -expand serialnumber
$newname = $serial + "-21"
Add-Computer -DomainName "cityhigh.lan" -OUPath 'OU=2021,OU=Laptops,DC=cityhigh,DC=lan' -NewName $newname -Restart 

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Yeah, didn't get any red errors

oBda

Using your script, it tells me the new name is no different from the current name

It seems that $newname is NOT adding the "-21" to the $serial

Same computer as you've just tried? If so, you may find it's just waiting for a reboot to complete the rename.

If you haven't rebooted since your Rename-Computer, you need to do that now. A rename requires a reboot before it's fully active.
You can try to rename it to its former name before the reboot:

Rename-Computer -NewName OldName -Force

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Still no go.  Maybe you can't do both at once??

"no go" is not an error report based on which we can help you.
I can't test it at the moment, but I'm pretty certain that Microsoft didn't add the -NewName argument to Add-Computer just to deceive you.

Sorry....but a restart didn't help or change the name.  The error is still saying the new name is the same as the current name.

Try a third name, just to see if that breaks the pattern a bit.

Well - is it?
And be aware that a Windows computer name is limited to 15 characters; if the serial + "21" is longer, the command might just cut off the rest.

$newname = "$(Get-WmiObject win32_bios | select -expand serialnumber)-21"
"This computer is currently named: '$(& hostname.exe)'"
"The new name based on the serial: '$($newname)'"

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Well, oBdA, your script to show current and new name worked great...it showed the current and correct new name

However, running that along with Add-Computer, didn't work....it added to the domain, but did not change the name

And no red errors either.

This is what I just ran

$newname = "$(Get-WmiObject win32_bios | select -expand serialnumber)-21"
Rename-Computer -NewName $newname -Force
Add-Computer -DomainName "cityhigh.lan" -OUPath "OU=2021,OU=Laptops,DC=cityhigh,DC=lan" -Restart

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Hat's off to you....that did it!  Thanks so much