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cls
[xml]$a = Get-Content  ("D:\Dropbox\temp\rechnernetz.xml")
$n = $a.SelectNodes("//pc")
foreach ($p in $n) {
    if ($p.name -eq "Franks Notebook") {
        $a.rechnernetz.RemoveChild($p)
    }
}
$a.Save("D:\Dropbox\Temp\rechnernetz_neu.xml")

<rechnernetz subnet="255.255.255.0" >
  <pc mac="23:cd:11:f1:2a:22">
		<ip>192.168.178.2</ip>
		<name>Franks Desktop</name>
	</pc>
	<pc mac="23:cd:11:f1:2a:23">
		<ip>192.168.178.3</ip>
		<name>Franks Notebook</name>
	</pc>
	<router mac="23:cd:11:f1:2a:24">
		<ip>192.168.178.1</ip>
		<name>Fritzbox</name>
	</router>
	<nas mac="23:cd:11:f1:2a:33">
		<ip>192.168.178.5</ip>
		<name>MP3 Server</name>
	</nas>
	<nas mac="23:cd:11:f1:2a:34">
		<ip>192.168.178.6</ip>
		<name>Spielfilme</name>
	</nas>
</rechnernetz>