Been thinking about this off and on as I want to give edit + users the ability to create short urls (and you can't at the moment)
If you have full access to Matrix the best option as outlined above is the remap manager or the webpaths screen of the asset.
If you wanted to automate it though you can use 'redirect page' assets - and manipulate their location in Matrix to generate short urls.
You could create a redirect asset under the site root so it gets a nice url EG: www.mysite.com/nice_webpath
Then run a trigger to new link it it into a folder elsewhere in the site and delete all other links - EG: /admin/remaps
Matrix would then auto add a remap from the nice url to the new url
so www.mysite.com/nice_webpath would remap to www.mysite.com/admin/remaps/nice_webpath and then to the asset specified in the "redirect" asset.
Of course it would be much cleaner if there were a trigger action - or something in the APIs.