JettyGwtSpringSample web application
Just some generic note about the code:


If the parameter called input contains a name you should rename it to name to express its purpose.
Method names should be verbs. I would rename greetServer to serveGreeting or doGreeting. (Check Code Conventions for the Java Programming Language, Naming Conventions and Clean Code, page 25.)
Comments like this are rather just noise:        

// If the input is not valid, throw an IllegalArgumentException back to
// the client.


It's pretty obvious from the code itself, so I'd remove the comment. (Clean Code by Robert C. Martin: Chapter 4: Comments, Noise Comments)
I'd move the complete validation logic to the FieldVerifier class (or another helper class or helper method) for better reusability and higher abstraction. It also would make the greetServer method more compact.
The isValidName could be a little bit shorter with the null-safe Apache Commons Lang StringUtils.length:

public static boolean isValidName(String name) {
    return StringUtils.length(name) &gt; 3;
}

Usage of Google Guava would make checkName more compact:

public static void checkName(final String name) {
    checkArgument(isValidName(name), 
        "Name must be at least 4 characters long");
}


(See also: Effective Java, 2nd edition, Item 47: Know and use the libraries The author mentions only the JDK's built-in libraries but I think the reasoning could be true for other libraries too.)
public String greetServer(String name) throws IllegalArgumentException {
    ...
}


IllegalArgumentException is a RuntimeException, so declaring them in the method signature is not mandatory, you could omit it:

public String greetServer(String name) {
    ...
}