Lowercase the second match in a combination of words using Regex.Replace

You can achieve your expected output using

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.RegularExpressions;
using System.Globalization;

public class Test
{
    public static void Main()
    {
        var strings = new List<string> { "Van Den Broek", "Derksen-van 't schip", "In Het Lid-Van De Boer"};
        var textInfo = new CultureInfo("en-US", false).TextInfo;
        var pattern = new Regex(@"\b(Van|Den|Der|In|de|het)\b(?:\s+(\w+))?", RegexOptions.Compiled|RegexOptions.IgnoreCase);
        foreach (var s in strings)
            Console.WriteLine(pattern.Replace(s, m => textInfo.ToTitleCase(m.Groups[1].Value) + 
               (m.Groups[2].Success ? $" {m.Groups[2].Value.ToLower()}" : "")));
    }
}

See the online demo yiedling

Van den Broek
Derksen-Van 't schip
In het Lid-Van de Boer

The \b(Van|Den|Der|In|de|het)\b(?:\s+(\w+))? regex matches a word from the Van, Den, Der, In, de and het list capturing it into Group 1, and then an optional sequence of one or more whitespaces and then any word captured into Group 2.

The match is replaced with Group 1 turned to title case (note the use of System.Globalization.ToTitleCase) and if Group 2 matched, a space and Group 2 value turned to lower case.