Application for passport in Hong Kong

Hi, for HK passport renewal, you can submit her application to Chinese consulate in New York, or mail it direct to the Immigration Department in Hong Kong. Please note that for the latter option, they require the originals of the supporting documents and collect your passport in person in Hong Kong. Or, you can apply for her passport renewal online if your daughter is aged 11 or above and will able to collect your new passport in person in HK.
Documents needed are a recent photograph, a copy of her Hong Kong permanent identity card (PIC) and other supporting document (if any), and the application form ID 843.
About the payment, you need to fill a payment form (ID 853) and a cheque or bank draft made payable to 'The Government of the Hong Kong Special Administrative Region' in HK dollars or US dollars. For more information, you can refer to http://www.gov.hk/en/residents/immigration/traveldoc/hksarpassport/overseasapplication.htm