The best way to do this is in version 4.2+ which allows using the aggregation pipeline in the update document and the updateOne, updateMany, or update(deprecated in most if not all languages drivers) collection methods.

MongoDB 4.2+

Version 4.2 also introduced the $set pipeline stage operator, which is an alias for $addFields. I will use $set here as it maps with what we are trying to achieve.

db.collection.<update method>(
    {},
    [
        {"$set": {"name": { "$concat": ["$firstName", " ", "$lastName"]}}}
    ]
)

Note that square brackets in the second argument to the method specify an aggregation pipeline instead of a plain update document because using a simple document will not work correctly.

MongoDB 3.4+

In 3.4+, you can use $addFields and the $out aggregation pipeline operators.

db.collection.aggregate(
    [
        { "$addFields": { 
            "name": { "$concat": [ "$firstName", " ", "$lastName" ] } 
        }},
        { "$out": <output collection name> }
    ]
)

Note that this does not update your collection but instead replaces the existing collection or creates a new one. Also, for update operations that require “typecasting”, you will need client-side processing, and depending on the operation, you may need to use the find() method instead of the .aggreate() method.

MongoDB 3.2 and 3.0

The way we do this is by $projecting our documents and using the $concat string aggregation operator to return the concatenated string.
You then iterate the cursor and use the $set update operator to add the new field to your documents using bulk operations for maximum efficiency.

Aggregation query:

var cursor = db.collection.aggregate([ 
    { "$project":  { 
        "name": { "$concat": [ "$firstName", " ", "$lastName" ] } 
    }}
])

MongoDB 3.2 or newer

You need to use the bulkWrite method.

var requests = [];
cursor.forEach(document => { 
    requests.push( { 
        'updateOne': {
            'filter': { '_id': document._id },
            'update': { '$set': { 'name': document.name } }
        }
    });
    if (requests.length === 500) {
        //Execute per 500 operations and re-init
        db.collection.bulkWrite(requests);
        requests = [];
    }
});

if(requests.length > 0) {
     db.collection.bulkWrite(requests);
}

MongoDB 2.6 and 3.0

From this version, you need to use the now deprecated Bulk API and its associated methods.

var bulk = db.collection.initializeUnorderedBulkOp();
var count = 0;

cursor.snapshot().forEach(function(document) { 
    bulk.find({ '_id': document._id }).updateOne( {
        '$set': { 'name': document.name }
    });
    count++;
    if(count%500 === 0) {
        // Excecute per 500 operations and re-init
        bulk.execute();
        bulk = db.collection.initializeUnorderedBulkOp();
    }
})

// clean up queues
if(count > 0) {
    bulk.execute();
}

MongoDB 2.4

cursor["result"].forEach(function(document) {
    db.collection.update(
        { "_id": document._id }, 
        { "$set": { "name": document.name } }
    );
})