How to modify binding-path at WPF application runtime

Due to the lack of context, let's try to establish one.

Assuming your ListBox is binding to a Person list with FirstName and LastName property, initial binding would be FirstName and you want to change it to LastName during runtime when clicking on a button.

This is how you can achieve it.

View

<ListBox Name="LstBx" ItemsSource="{Binding PersonList}">
    <ListBox.ItemTemplate>
        <DataTemplate>
            <TextBox Name="tb1" Text="{Binding Path=FirstName}"/>
        </DataTemplate>
    </ListBox.ItemTemplate>
</ListBox>

<Button Click="Button_Click" Width="100" Height="20" Content="Change Binding"/>

CodeBehind

public List<Person> PersonList { get; set; } = new List<Person>
{
    new Person { FirstName = "ABC", LastName = "123" },
    new Person { FirstName = "DEF", LastName = "456" },
    new Person { FirstName = "GHI", LastName = "789" }
};

public MainWindow()
{
    InitializeComponent();
    DataContext = this;
}

private void Button_Click(object sender, RoutedEventArgs e)
{
    foreach (var person in PersonList)
    {
        var listBoxItem = LstBx.ItemContainerGenerator.ContainerFromItem(person);
        var contentPresenter = FindVisualChild<ContentPresenter>(listBoxItem);
        var target = contentPresenter.ContentTemplate.FindName("tb1", contentPresenter) as TextBox;

        if (target != null)
        {
            var binding = new Binding
            {
                // Remember each ListBoxItem's DataContext is the individual item 
                // in the list, not the list itself.
                Source = person, 
                Path = new PropertyPath(nameof(Person.LastName)),
                // Depends on what you need.
                //Mode = BindingMode.TwoWay,
                //UpdateSourceTrigger = UpdateSourceTrigger.PropertyChanged
            };

            BindingOperations.SetBinding(target, TextBox.TextProperty, binding);
        }
    }
}

// Available from MSDN
private Child FindVisualChild<Child>(DependencyObject obj) where Child : DependencyObject
{
    for (int i = 0; i < VisualTreeHelper.GetChildrenCount(obj); i++)
    {
        var child = VisualTreeHelper.GetChild(obj, i);
        if (child != null && child is Child)
        {
            return (Child)child;
        }
        else
        {
            var childOfChild = FindVisualChild<Child>(child);

            if (childOfChild != null) { return childOfChild; }
        }
    }
    return null;
}