Design pattern for "retrying" logic that failed?

You could try the Idempotent Retry Pattern.

I really like this Java 8 code from this blog and you don't need any extra library on your classpath.

You only need to pass a function to the retry class.

@Slf4j
public class RetryCommand<T> {

    private int maxRetries;

    RetryCommand(int maxRetries)
    {
        this.maxRetries = maxRetries;
    }

    // Takes a function and executes it, if fails, passes the function to the retry command
    public T run(Supplier<T> function) {
        try {
            return function.get();
        } catch (Exception e) {
            log.error("FAILED - Command failed, will be retried " + maxRetries + " times.");
            return retry(function);
        }
    }

    private T retry(Supplier<T> function) throws RuntimeException {

        int retryCounter = 0;
        while (retryCounter < maxRetries) {
            try {
                return function.get();
            } catch (Exception ex) {
                retryCounter++;
                log.error("FAILED - Command failed on retry " + retryCounter + " of " + maxRetries, ex);
                if (retryCounter >= maxRetries) {
                    log.error("Max retries exceeded.");
                    break;
                }
            }
        }
        throw new RuntimeException("Command failed on all of " + maxRetries + " retries");
    }
}

And to use it:

new RetryCommand<>(5).run(() -> client.getThatThing(id));

Shameless plug: I have implemented some classes to allow retrying operations. The library is not made available yet, but you may fork it on github. And a fork exists.

It allows building a Retryer with various flexible strategies. For example:

Retryer retryer = 
    RetryerBuilder.newBuilder()
                  .withWaitStrategy(WaitStrategies.fixedWait(1, TimeUnit.SECOND))
                  .withStopStrategy(StopStrategies.stopAfterAttempt(3))
                  .retryIfExceptionOfType(IOException.class)
                  .build();

And you can then execute a callable (or several ones) with the Retryer:

retryer.call(new Callable<Void>() {
    public Void call() throws IOException {
        connection = newConnection();
        return null;
    }
}

Using Failsafe (author here):

RetryPolicy retryPolicy = new RetryPolicy()
  .retryOn(IOException.class)
  .withMaxRetries(5)
  .withDelay(1, TimeUnit.SECONDS);

Failsafe.with(retryPolicy).run(() -> newConnection());

No annotations, no magic, doesn't need to be a Spring app, etc. Just straightforward and simple.