Check out the context of your function. It has property context.logStreamName. You could change it to any unique name.

Check more details in the next article - The Context Object (Node.js)

EDIT: Don't pay attention to my previous answer. It was completely wrong.

I checked my code how I implemented the same functionality. It isn't possible to change group or stream name from the context. But you could use CloudWatchLogs in your code for putting logs in specific group and stream.

My code:

var AWS = require('aws-sdk');

exports.handler = (event, context, callback) => {
    // TODO implement
    var cloudwatchlogs = new AWS.CloudWatchLogs({ apiVersion: '2014-03-28' });

    // create new stream name for each request
    // because I don't persist sequenceToken
    var logStreamName = "myStreamName" + Math.random()

    var params = {
        logGroupName: 'myLogGroup',
        logStreamName: logStreamName
    };
    cloudwatchlogs.createLogStream(params, function(err, data) {
        if (err) console.log(err, err.stack); // an error occurred
        else {
            var params = {
                logEvents: [{
                        message: 'log message',
                        timestamp: new Date().getTime()
                    },
                    {
                        message: 'one more log message',
                        timestamp: new Date().getTime()
                    }
                ],
                logGroupName: 'myLogGroup',
                logStreamName: logStreamName
            };

            cloudwatchlogs.putLogEvents(params, function(err, data) {
                if (err) console.log(err, err.stack); // an error occurred
                else console.log(data); // successful response
            });
        }
    });
    callback(null, 'Hello from Lambda');
};

Disadvantages of this approach are:

• Your lambda creates two logGroups and two logStreams in accordance. One of your lambda, another custom. console.log('') writes to lambda log, putLogEvents writes to your custom log (i.e: myStreamName0.10141409975385463). Be aware of this.
• You have to create manually 'myLogGroup' (how I did) or implement code that will create a logGroup if it doesn't exist.

The result of this testing code looks like:

My productions stream names have more sense rather name + random. And they are using guid or timestamp as a suffix. I don't think that it is safe to use random.

You cannot change the Log Stream name. Each running container for your Lambda function logs into a different Log Stream.

If you want to easily see the logs together as one, you can stream them to Amazon ElasticSearch Service so you can use Kibana to browse through your logs or you can use CLI tools like awscli to aggregate and browse them in your terminal.

If you want to keep track of the executions anyway like in my case there is also a less redundant approach:

Just create a dynamodb table "Executions" and make sure to push a new entry on every lambda execution with both the logStreamName (which can be accessed from context) and your custom name or other custom information that you need to look up this logStream.

Upsides:

• no wrapper/custom logging logic necessary.
• no duplicate entries in cloudwatch

Downsides:

• you might need to do an additional call to dynamodb to find out the logStreamName as the logStreamName itself has no identifying value
• no considerations about IAM and access so far. you have to take that in mind in your setup