You will need a custom module that only changes the frontname of a module (
Mage_Customer in this case.)
Let's name the module
Easylife_UserThis module contains only 2 files.
app/etc/module/Easylife_User.xml - the declaration file
<Mage_Customer /><!-- should depend on Mage_Customer so it is loaded after it -->
app/code/local/Easylife/User/etc/config.xml - the configuration file
<customer>/user/</customer><!-- might be needed if you have secure urls but not sure -->
<customer><!-- for the customer route... -->
<frontName>user</frontName><!-- just change the front name -->
Clear the cache and you are done.
If all your links to the customer pages are generated using
->getUrl('customer/...'). This should work.
If you have links in your templates like this
It won't work. But anyway, you shouldn't have links like the example above.