Can a child class implement the same interface as its parent?

Yes, you can explicitly redeclare that you want to implement IDispatch, and implement it explicitly again in Bar.

However, you won't be able to call the original implementation in Foo. If you need to do that, you'll need to change Foo either to use implicit interface implementation with a virtual method (which can be overridden and then called with base.Dispatch() in Bar) or make the Foo implementation call a protected virtual method which again you'd override in Bar.

You could, alternatively, do this one of two ways:

First, don't implement the interface explicitly:

public class Foo : IDispatch {
    public virtual void Dispatch() {
        whatever();
    }
}

public class Bar : Foo {
    public override void Dispatch() {
        whateverElse();
    }
}

Second, implement it explicitly but add a function that the child class can override:

public class Foo : IDispatch {
    void IDispatch.Dispatch() {
        this.Dispatch();
    }

    protected virtual void Dispatch() {
        whatever();
    }
}

public class Bar : Foo {
    protected override void Dispatch() {
        whateverElse();
    }
}