If you are correctly coding to the JDBC spec, and not using database-specific functionality, you should not need to include the MySQL JDBC driver JAR file in your project - the JDBC API is all in the JDK runtime JAR.
This sounds like the classical confusion between the IDE and the deployable application.
The ONLY time you need to put ANY database driver "in Eclipse" is if you have an Eclipse tool such as one of the database management plug-ins and you need to tell the plug-in where to find the driver.
For general app development, Eclipse could care less (and in fact, has no use for databases). In that case, what you really want is to make the driver available to the application. And, specifically, if Eclipse is managing a web application server for debugging purposes, to that application server environment. And, as in general databases, Eclipse doesn't know nor does it care about web application servers. Only an installed plugin for webapp development such as sysdeo or WTP will care.
Normally, you should not put application libraries in your webapp server's private libraries, you should put them in the webapp's WEB-INF/lib directory. One of the few exceptions is JDBC drivers. Normally, you'd want to place things like the mysql connector jar in the server's common library directory (for example, CATALINA_HOME/lib, for Tomcat). That's because unlike most Java code, the drivers are intended to be shared between multiple applications, and may, in fact, do better resource management if they do. It also insulates your webapps from dependence on a particular driver/version and makes them somewhat more portable.
An IDE is no substitute for an Intelligent Developer.
When the OP mentioned an Eclipse error, I assumed that it was a compile error. The OP was not clear on this topic. As Tim pointed out, there is a big difference between building an app that uses JDBC and running it - you don't need the driver to build it.