Change default icon for moving cells in UITableView

This is a really hacky solution, and may not work long term, but may give you a starting point. The re-order control is a UITableViewCellReorderControl, but that's a private class, so you can't access it directly. However, you could just look through the hierarchy of subviews and find its imageView.

You can do this by subclassing UITableViewCell and overriding its setEditing:animated: method as follows:

- (void) setEditing:(BOOL)editing animated:(BOOL)animated
{
    [super setEditing: editing animated: YES];

    if (editing) {

        for (UIView * view in self.subviews) {
            if ([NSStringFromClass([view class]) rangeOfString: @"Reorder"].location != NSNotFound) {
                for (UIView * subview in view.subviews) {
                    if ([subview isKindOfClass: [UIImageView class]]) {
                        ((UIImageView *)subview).image = [UIImage imageNamed: @"yourimage.png"];
                    }
                }
            }
        }
    }   
}

---

Or in Swift

override func setEditing(_ editing: Bool, animated: Bool) {
    super.setEditing(editing, animated: animated)

    if editing {
        for view in subviews where view.description.contains("Reorder") {
            for case let subview as UIImageView in view.subviews {
                subview.image = UIImage(named: "yourimage.png")
            }
        }
    }
}

Be warned though... this may not be a long term solution, as Apple could change the view hierarchy at any time.

Swift version of Rick's answer with few improvements:

override func setEditing(editing: Bool, animated: Bool) {
    super.setEditing(editing, animated: animated)

    if editing {
        if let reorderView = findReorderViewInView(self), 
            imageView = reorderView.subviews.filter({ $0 is UIImageView }).first as? UIImageView {
            imageView.image = UIImage(named: "yourImage")
        }
    }
}

func findReorderViewInView(view: UIView) -> UIView? {
    for subview in view.subviews {
        if String(subview).rangeOfString("Reorder") != nil {
            return subview
        }
        else {
            findReorderViewInView(subview)
        }
    }
    return nil
}

Swift 4

   // Change default icon (hamburger) for moving cells in UITableView
    func tableView(_ tableView: UITableView, willDisplay cell: UITableViewCell, forRowAt indexPath: IndexPath) {
        let imageView = cell.subviews.first(where: { $0.description.contains("Reorder") })?.subviews.first(where: { $0 is UIImageView }) as? UIImageView

        imageView?.image = #imageLiteral(resourceName: "new_hamburger_icon") // give here your's new image
        imageView?.contentMode = .center

        imageView?.frame.size.width = cell.bounds.height
        imageView?.frame.size.height = cell.bounds.height
    }

Ashley Mills' answer was excellent at the time it was offered, but as others have noted in the comments, the view hierarchy has changed from version to version of iOS. In order to properly find the reorder control, I'm using an approach that traverses the entire view hierarchy; hopefully this will give the approach an opportunity to continue working even if Apple changes the view hierarchy.

Here's the code I'm using to find the reorder control:

-(UIView *) findReorderView:(UIView *) view
{
    UIView *reorderView = nil;
    for (UIView *subview in view.subviews)
    {
        if ([[[subview class] description] rangeOfString:@"Reorder"].location != NSNotFound)
        {
            reorderView = subview;
            break;
        }
        else
        {
            reorderView = [self findReorderView:subview];
            if (reorderView != nil)
            {
                break;
            }
        }
    }
    return reorderView;
}

And here's the code I'm using to override the -(void) setEditing:animated: method in my subclass:

-(void) setEditing:(BOOL)editing animated:(BOOL)animated
{
    [super setEditing:editing animated:animated];
    if (editing)
    {
        // I'm assuming the findReorderView method noted above is either
        // in the code for your subclassed UITableViewCell, or defined
        // in a category for UIView somewhere
        UIView *reorderView = [self findReorderView:self];
        if (reorderView)
        {
            // I'm setting the background color of the control
            // to match my cell's background color
            // you might need to do this if you override the
            // default background color for the cell
            reorderView.backgroundColor = self.contentView.backgroundColor;
            for (UIView *sv in reorderView.subviews)
            {
                // now we find the UIImageView for the reorder control
                if ([sv isKindOfClass:[UIImageView class]])
                {
                    // and replace it with the image we want
                    ((UIImageView *)sv).image = [UIImage imageNamed:@"yourImage.png"];
                    // note:  I have had to manually change the image's frame
                    // size to get it to display correctly
                    // also, for me the origin of the frame doesn't seem to
                    // matter, because the reorder control will center it
                    sv.frame = CGRectMake(0, 0, 48.0, 48.0);
                }
            }
        }
    }
}