BsonDocument has two internal fields for its implementation:

private List<BsonElement> _elements;
private Dictionary<string, int> _indexes; // maps names to positions

Considering that for small documents a linear search of the _elements List can be faster than a Dictionary lookup, we could postpone creating the dictionary until the document grew large enough to make it worthwhile, and fall back to linear searches when the dictionary doesn't yet exist. This would both save memory and be faster for smaller documents.

We would have to experiment to find the threshold at which to create the dictionary. A similar class in .NET (HybridDictionary) uses a threshold of 8, so that gives us an idea of the approximate value for the threshold.