[vasilis]

You must edit the /mail/html/order_invoice.tpl file (which will be either in skin/common_files dir or skin/<skin_used> dir if you have done any modifications already to that template).
Find the following code (around line 77 in 4.4.2 version):

<table cellspacing="0"{if $is_nomail eq 'Y'} class="invoice-personal-info"{else} cellpadding="0" style="width:45%; border: 0px none; margin-bottom: 15px;"{/if} summary="{$lng.lbl_address|escape}">
        
        <tr>
          <td><strong>{$lng.lbl_email}:</strong></td>
          <td>{$order.email}</td>
        </tr>
                        
        {if $_userinfo.default_fields.title}
          <tr>
            <td><strong>{$lng.lbl_title}:</strong></td>
            <td>{$order.title}</td>
          </tr>
        {/if}

        {if $_userinfo.default_fields.firstname}
          <tr>
            <td{if $is_nomail ne 'Y'} nowrap="nowrap"{/if}><strong>{$lng.lbl_first_name}:</strong></td>
            <td>{$order.firstname}</td>
          </tr>
        {/if}
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and wherever you want to display the username field, among the <tr> tags, insert an additional table row:

<tr>
          <td><strong>username:</strong></td>
          <td>{$userinfo.username}</td>
 </tr>

You might have to use different smarty variable names (although I don't think so), I've checked those in 4.4.2 version - check smarty variables in Webmaster mode.