Runnable with a parameter?

You could put it in a function.

String paramStr = "a parameter";
Runnable myRunnable = createRunnable(paramStr);

private Runnable createRunnable(final String paramStr){

    Runnable aRunnable = new Runnable(){
        public void run(){
            someFunc(paramStr);
        }
    };

    return aRunnable;

}

(When I used this, my parameter was an integer ID, which I used to make a hashmap of ID --> myRunnables. That way, I can use the hashmap to post/remove different myRunnable objects in a handler.)

theView.post(new Runnable() {
    String str;
    @Override                            
    public void run() {
        par.Log(str);                              
    }
    public Runnable init(String pstr) {
        this.str=pstr;
        return(this);
    }
}.init(str));

Create init function that returns object itself and initialize parameters with it.

Well it's been almost 9 years since I originally posted this and to be honest, Java has made a couple improvements since then. I'll leave my original answer below, but there's no need for people to do what is in it. 9 years ago, during code review I would have questioned why they did it and maybe approved it, maybe not. With modern lambdas available, it's irresponsible to have such a highly voted answer recommending an antiquated approach (that, in all fairness, was dubious to begin with...) In modern Java, that code review would be immediately rejected, and this would be suggested:

void foo(final String str) {
    Thread t = new Thread(() -> someFunc(str));
    t.start();
}

As before, details like handling that thread in a meaningful way is left as an exercise to the reader. But to put it bluntly, if you're afraid of using lambdas, you should be even more afraid of multi-threaded systems.

Original answer, just because:

You can declare a class right in the method

void Foo(String str) {
    class OneShotTask implements Runnable {
        String str;
        OneShotTask(String s) { str = s; }
        public void run() {
            someFunc(str);
        }
    }
    Thread t = new Thread(new OneShotTask(str));
    t.start();
}