Data stores used in serverless/Lambda applications are the same as any other application. The store that you choose will probably depend more on the data and the way you intend to use it than the fact that it's being accessed by Lambda - at the end of the day a client is a client, regardless of if it's serverless or not.
Some popular options for Lambda-based applications are:
DyanmoDB: Amazon's key/value store (i.e. it's NOT a relational database, so don't treat it like that) which is nice because it is also serverless - you just provision your table, and that's it.
S3: Amazon's object store, which can be used like a database in a surprising number of situations. Just remember that it is not a database, and is more of a key/value store (with different performance characteristics to DDB).
RDS: Amazon's MySQL/PgSql/MSSQL/etc service. RDS works well when you want/know/love relational databases, especially since it can easily be run with cross-AZ redundancy (meaning an outage won't affect your database or your serverless application). Note that your Lambda functions will need to be VPC-based in order to securely access your RDS instance.