I have a script that copies several large files over the network and I want
to track the performance, so I decided to use Measure-Command. How can I
store all the results in a simple csv file with the file name so I can tell
exactly how long each one took?

You can create a custom object with two fields --Filename and Duration-- for
each copied file and export the collection to a CSV file.

$path = # <files' location>
$dest = # <files' destination>
Get-ChildItem $path | ForEach-Object {
New-Object PSObject -Property @{
File = $_.Fullname
Duration = Measure-Command {Copy-Item $_.Fullname $dest}
}
} | Select-Object File, Duration | Export-Csv Perf.csv -NTI

--
Robert