First, you need to have pulled that field as part of your project metadata through the Add/Remove Components interface (under right click project->Force.com).
When the field metadata has been pulled, you'll have a file called
Contact.object in your
src/objects directory. Open that file and you'll find, as one portion of the lengthy XML source that defines the object, a number of entries that look like this:
You can alter your field's label there. Save the file in Eclipse and it should deploy automatically back to the server, unless you've configured your project to disable that behavior. Be aware, if you dive in, that it's fairly easy to introduce problems with dependencies by making data model changes in the XML source, and you'll get a deployment error if that happens. If you stick to field labels, you should be OK.
Note that you actually can change field API names, although of course you cannot change the API names of standard fields.