To tackle the reducing power system inertia, new requirements for Frequency Containment Reserves (FCR) were implemented in the Nordic Synchronous Area. Hydropower units, which dominate the FCR markets, demonstrate difficulties meeting these requirements due to limited power response speed. A cost-effective enhancement of the FCR power response is achieved by integrating an energy storage system as a secondary power source. This paper addresses an identified knowledge gap in the optimization possibilities of the storage system enhancing the FCR response: the selection of the optimal storage technology and storage system sizing. An optimization method is presented, minimizing the storage system life-cycle costs and returning the optimal control parameters and power and energy ratings. The use of three fast-response storage technologies is compared: supercapacitors, flywheels, and lithium-ion batteries. The method is validated on a reference hydropower unit model and tested on three existing units. Results indicate supercapacitors and flywheels are competitive technologies to batteries due to the low energy and high cycling demands, with supercapacitors being advantageous for higher frequency contributions. The obtained storage system costs represent a fraction of the potential added yearly FCR income, highlighting the economic viability of the enhancement.