You can use a Dialog form instead of using a message box.
Add a ContextMenuStrip item to the Dialog then add a click method to it.
Set the copy to clipborad to the text or label you want to be sent to the clipbboard. something like below.
Private Sub CopyMessageToolStripMenuItem_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles CopyMessageToolStripMenuItem.Click
Dim message As String
message = Label1.Text
My.Computer.Clipboard.Clear()
My.Computer.Clipboard.SetText(message)
End Sub