Remove specific array element by cell text string value [swift 4]

Avoid using 2 array to "persist" your models. Instead you can generate a single Array with tuples :

var myArray: [(String, Bool)] = [("Test", false), ("Test1", false), ("Test2", false)]

Starting here the problem is simplified, and you will not have index path issue again

Edit

I've changed my code to support [ListItem] saving to UserDefaults- that comment brought by Leo Dabus I also changed a couple of lines that were inspired by vadian's code who appear to have a great coding style.

class ChecklistViewController: BaseViewController, UITableViewDelegate, UITableViewDataSource{

    var dataHolder: [ListItem] = DefaultsHelper.savedItems

    @IBOutlet weak var myTableView: UITableView!


    func tableView(_ tableView: UITableView, numberOfRowsInSection section: Int) -> Int {
        return dataHolder.count
    }

    func tableView(_ tableView: UITableView, cellForRowAt indexPath: IndexPath) -> UITableViewCell {

        let cell = UITableViewCell(style: UITableViewCell.CellStyle.default, reuseIdentifier: "cell")

        cell.textLabel?.font = UIFont.boldSystemFont(ofSize: 18.0)

        let currentListItem = dataHolder[indexPath.row]

        cell.textLabel?.text = currentListItem.title
        cell.accessoryType = currentListItem.isChecked ? .checkmark : .none

        return cell
    }

    func tableView(_ tableView: UITableView, didSelectRowAt indexPath: IndexPath) {

        dataHolder[indexPath.row].isChecked.toggle()
        DefaultsHelper.saveItems(items: dataHolder)

        tableView.reloadRows(at: [indexPath], with: .none)
        tableView.deselectRow(at: indexPath, animated: true)
    }

    func tableView(_ tableView: UITableView, commit editingStyle: UITableViewCell.EditingStyle, forRowAt indexPath: IndexPath) {

        if editingStyle == .delete {

            dataHolder.remove(at: indexPath.row)
            DefaultsHelper.saveItems(items: dataHolder)

            myTableView.reloadData()
            myTableView.deleteRows(at: [indexPath], with: .automatic)
        }
    }


    override func viewDidAppear(_ animated: Bool) {
        super.viewDidAppear(animated)

        myTableView.reloadData()
    }

    override func viewDidLoad() {
        super.viewDidLoad()

        // be sure you've set your tableView's dataSource and delegate to this class (It's fine if you've handled this on the storyboard side)

        addSlideMenuButton()
    }

    @IBAction func addNewObject(_ sender: Any) {

        let alert = UIAlertController(title: "New Item", message: nil, preferredStyle: .alert)
        alert.addTextField { (alertInputTextField) in
            alertInputTextField.autocapitalizationType = .sentences
        }

        alert.addAction(UIAlertAction(title: "Cancel", style: .default, handler: { (action) in
            self.dismiss(animated: true, completion: nil)
        }))

        alert.addAction(UIAlertAction(title: "Add", style: .default, handler: { (action) in

            let textf = alert.textFields![0] as UITextField

            let indexPath = IndexPath(row: self.dataHolder.count, section: 0)

            let itemToInsert = ListItem(title: textf.text!, isChecked: false)

            // self.dataHolder.append(itemToInsert)

            // thought you would want this, it will add your notes in reverse chronological order
            self.dataHolder.insert(itemToInsert, at: 0)
            DefaultsHelper.saveItems(items: self.dataHolder)

            self.myTableView.insertRows(at: [indexPath], with: .automatic)
        }))

        self.present(alert, animated: true, completion: nil)

    }
}

Model classes:

// implementing NSObject and NSCoding to let us save this item in UserDefaults
class ListItem: NSObject, NSCoding{

    var title: String
    var isChecked: Bool

    init(title: String, isChecked: Bool) {
        self.title = title
        self.isChecked = isChecked
    }

    // This code lets us save our custom object in UserDefaults

    required convenience init(coder aDecoder: NSCoder) {
        let title = aDecoder.decodeObject(forKey: "title") as? String ?? ""
        let isChecked = aDecoder.decodeBool(forKey: "isChecked")
        self.init(title: title, isChecked: isChecked)
    }

    func encode(with aCoder: NSCoder) {
        aCoder.encode(title, forKey: "title")
        aCoder.encode(isChecked, forKey: "isChecked")
    }
}

class DefaultsHelper{

    private static let userDefaults = UserDefaults.standard
    private static let dataKey = "dataHolder"

    static var savedItems: [ListItem] {
        guard let savedData = userDefaults.data(forKey: dataKey) else { return [] }

        do{
            let decodedData = try NSKeyedUnarchiver.unarchiveTopLevelObjectWithData(savedData)
            return decodedData as? [ListItem] ?? []
        }catch{
            print("could not fetch items- you may handle this", error)
        }

        return []
    }

    static func saveItems(items: [ListItem]){
        do{
            let encodedData = try NSKeyedArchiver.archivedData(withRootObject: items, requiringSecureCoding: false)
            userDefaults.set(encodedData, forKey: dataKey)
        }catch{
            print("could not save items- you may handle this", error)
        }
    }
}