The Italian National Institute for Astrophysics (INAF) is leading the ASTRI project within the ambitious Cherenkov Telescope Array (CTA), the next generation of ground-based observatories for very high energy gamma-ray astronomy. In the framework of the small sized telescopes (SST), a first goal of the ASTRI project is the realization of an end-to-end prototype in dual-mirror configuration (2M) with the camera composed of a matrix of Silicon photo-multiplier sensors managed by innovative front-end and back-end electronics. The prototype, named ASTRI SST-2M, is installed in Italy at the INAF “M.G. Fracastoro” observing station located at Serra La Nave, 1735 m a.s.l. on Mount Etna, Sicily. As a second step, the ASTRI project is focused on the implementation of a mini-array composed at least of nine ASTRI telescopes and proposed to be placed at the CTA southern site. This paper outlines the design of the camera server software that will be installed on the ASTRI mini-array. The software is based on the version installed on the ASTRI SST-2M prototype operating in a single telescope configuration. The migration from single telescope to mini-array context has required additional interfaces in order to guarantee high interoperability with other software and hardware components. In the mini-array configuration each camera communicates with its own camera server via a dedicated high rate data link. The primary goal of the camera server is to acquire the bulk data, packet by packet, without any data loss and to timestamp each packet very precisely. During array operation, the camera server receives from the SoftWare Array Trigger (SWAT) the list of science events that participate in stereo triggered events. These science events, and all others that are flagged either by the camera as interleaved calibration or by the camera server as possible single-muon events, are sent to the Array DAQ. All remaining science events will be discarded. A suitable buffer is provided to perform this processing on all the incoming event packets. The camera server provides interfaces to the array control software to allow for monitoring and control during array operations. In this paper we present the design of the camera server software with particular emphasis on the external interfaces. In addition, we report the results of the first integration activities and performance tests.