How to Write-Verbose from Invoke-Command?

Try it this way:

Invoke-Command -ComputerName MY-COMPUTERNAME  {$VerbosePreference='Continue'; Write-Verbose "blah" }

Best thing I think you could do is create a function that has [CmdletBinding()] so it supports the -verbose switch. Then you would be able to capture the verbose state of the local function using the local $VerbosePreference and pass it along in the invoke command. This will only work in Powershell 3.0 and higher, since you will need to use the $Using scope modifier.

Function write-blah{
[CmdletBinding()]
Param()
Invoke-Command -ComputerName MY-COMPUTERNAME  {$VerbosePreference=$Using:VerbosePreference; Write-Verbose "blah" }
}

Then you would call your function like this.

Write-Blah -verbose

In testing this has worked for me. I believe your function must support parameters, hence the empty Param() block.