Looking at psonesixmigrator-v2.1.1 classes/UpgraderTools.php file, you can try set "Server performance" setting to "Low" inside your current PS 1.6 store. If it is not sufficient, you can lower values inside mentioned file (classes/UpgraderTools.php; lines 244 - 252).
BTW, are you getting some errors in your hosting logs?
If nothing helps, I would make a store and database copy locally, and try update here. If not successful either, I would disable (or uninstall) any third party modules.