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Код

Console.WriteLine("--------- Вариант 1 ---------------");
var query2 =
    from parts in table2.AsEnumerable()
    select parts.Field<string>("name").ToLower();
var query1 = 
    from names in  table1.AsEnumerable()
    where query2.Any(part => names.Field<string>("name").ToLower().Contains(part))
    select names.Field<string>("name");

foreach (string name in query1)
    Console.WriteLine(name);

Console.WriteLine("--------- Вариант 2 ---------------");
var results = table1.AsEnumerable()
    .Where
    (
        row =>
        {
            if (table2.AsEnumerable().Any(part => row.Field<string>("name").ToLower().Contains(part.Field<string>("name").ToLower())))
                return true;
            else
                return false;
        }
    )
    .Select<DataRow, string>
    (
        row => row.Field<string>("name")
    );

foreach (string name in results)
    Console.WriteLine(name);

Console.WriteLine("--------- Вариант 3 ---------------");
var query3 =
    from names in table1.AsEnumerable()
    let nameSource = names.Field<string>("name").ToLower()
    from parts in table2.AsEnumerable()
    let partName = parts.Field<string>("name").ToLower()
    where nameSource.Contains(partName)
    select names.Field<string>("name");

foreach (string name in query3)
    Console.WriteLine(name);