There is actually already a field made for this in SharePoint. There is also a page where you can change the order as you wish. This is already used in URL lists if I'm not mistaken, but you could do the following:
Visit the order page
If you visit */_layouts/reorder.aspx?List={list id}*, you will see all the items in your list or library, and you can change their ordering.
Update the view
Now your items have an order but it's not used in your views. You can modify the view (for example, in SharePoint Designer) and change the 'OrderBy' property to 'Order'.
Unfortunately I don't know if you can just add the column to your list and use it in the GUI. It appears that this is a hidden column of some sort.
Add a ribbon button to the order page
This would make it much easier to modify the ordering in the future. You could do this via SharePoint Designer, or with an element file in a solution:
This page has more information about it: http://consultingblogs.emc.com/mattlally/archive/2012/01/16/add-change-item-order-to-a-custom-link-list.aspx