How to create custom menus with different path
If you want to give user view page then this :

1) You need to give path as user. That will take you to the currently logged-in user account.

Or if you want go to view edit page:

2) You can use menu token module to give path as user edit page.You can use the token replacements to get to the user edit page.Hope this helps you.

3) You can use user/[current-user:uid]/edit once you have the menu_token module installed.But i dont know why its not working.

You can try installing me aliases module and give path as user/me/edit. This will give option for editing user account.Hope this resolves your current issue.