The default way the sender showsup in emails from phpBB forum is “<forum@domain.com>”.
This mod makes it look like “Forum Name <forum@domain.com>”.

Open includes/functions_messenger.php

Find:
$headers[] = 'From: ' . $this->from;
Replace with:
$headers[] = 'From: "' . str_replace("&", "&", $config['sitename']) .'" ' . $this->from;
Find:
$headers[] = 'Sender: <' . $config['board_email'] . '>';
Replace with:
$headers[] = 'Sender: "' . str_replace("&amp;", "&", $config['sitename']) .'" <' . $config['board_email'] . '>';

Upload the modified file and you are done.

Note: Some special characters, in the forum name, may cause some problems with this mod.