Для драйвера c # 2.1 (MongoDB 3.0)
var collection = database.GetCollection<BsonDocument>("<<name of the collection>>");
var filter = Builders<BsonDocument>.Filter.Regex("name", new BsonRegularExpression("Joe"));
var result = await collection.Find(filter).ToListAsync();
Для драйвера c # 2.2 (MongoDB 3.0)
var filter = new BsonDocument { { parameterName, new BsonDocument { { "$regex", value }, { "$options", "i"} } } }
var result = collection.Find(filter).ToList();