I would not recommend creating a custom entity in place of Account. There are a number of special features such as marketing lists, campaigns, orders, and cases that are only designed to work only with accounts, contacts and (sometimes) leads. Moreover, many 3rd party add-ons rely on these entities.
Instead of replacing the account entity, take the approach that Jessamyn suggested and simply customize the Account entity to meet your needs. If the term "Account" is really an issue, it can be changed in most places by editing the display name in the entity definition and customizing the site map. You may have to change labels in some forms and views too.
Because we are a university, we thought about renaming the account entity, but in the end just left the "account" name alone. It hasn't been a problem for our users once they understand that "Account" = "Organization".