Hello, this is very simple, but yet useful modification 
It makes the 'Stay logged in' checkbox be checked as default
Do the following for the checkbox in the login bar:
Find in template: login_bar_form:
Replace with:
Do the following for the checkbox at the login form:
Find in template: helper_login_form:
Replace with:

It makes the 'Stay logged in' checkbox be checked as default

Do the following for the checkbox in the login bar:
Find in template: login_bar_form:
PHP:
<label for="ctrl_remember" class="rememberPassword"><input type="checkbox" name="remember" value="1" id="ctrl_remember" tabindex="104" /> {xen:phrase stay_logged_in}</label>
Replace with:
PHP:
<label for="ctrl_remember" class="rememberPassword"><input type="checkbox" name="remember" value="1" checked="checked" id="ctrl_remember" tabindex="104" /> {xen:phrase stay_logged_in}</label>
Do the following for the checkbox at the login form:
Find in template: helper_login_form:
PHP:
<div><label for="ctrl_pageLogin_remember" class="rememberPassword"><input type="checkbox" name="remember" value="1" id="ctrl_pageLogin_remember" /> {xen:phrase stay_logged_in}</label></div>
Replace with:
PHP:
<div><label for="ctrl_pageLogin_remember" class="rememberPassword"><input type="checkbox" name="remember" value="1" checked="checked" id="ctrl_pageLogin_remember" /> {xen:phrase stay_logged_in}</label></div>
Great little tweak. ... Why not stay logged in 