On receiving a advertisement packet/scan response packet event in the client stack handler, you can access the passed eventParam to get the data associated with the actual advertisement packet. Thus, if you include the name in the advertisement packet, then you will be able to read it quite easily.
Please note that you will (obviously) need to be scanning to receive the advertisement packets as well.
CYBLE_GAPC_ADV_REPORT_T * report = eventParam;
//Other code to use it here
EPratt is correct. If your CyBLE is a Central, and you want to get the device name embedded in adv packet, then you can read it using the method Epratt mentioned. But the problem is you have to know which all adv packet index contains the device name. Otherwise it will be difficult to find the adv packet start index of device name data.
Thanks for your help. Anjana - it looks to me like the <<Local Name>> data in the advertisement packet has a value of 9 so I've used this information as an initial check to make sure I'm looking in the correct place.