If you right click on the screen, then select Inspect, then find the console this will show you any Javascript errors. You can sit with this open while you type the name and see if anything gets reported.
for the PHP error log you need to add the following to your wp-config.php:
define('WP_DEBUG', true);
define('WP_DEBUG_LOG', true);
define('WP_DEBUG_DISPLAY', true);
@ini_set('display_errors', 0);
You can then load a plugin such as BES Error Log Viewer to view any errors recorded in this file.
Another useful plugin is Query Monitor, which gives you dynamic reports of errors on your system.
