Applications open today, 20th March, https://developers.google.com/open-source/gsoc/. On April 3 - 24 Organizations review and select student proposals. We will submit more technical details about this project next week, so that you can get a feeling how and where to start to forge LAN Sync into the ownCloud distributed system. There are certain requirements to be satisfied.
General idea here is that client should first search in its local network for missing new/updated files before it asks the server. This means, you might need to introduce some consensus algorithm to avoid conflicts between client and server (State at the client needs to be reflected on the server eventualy), so if you sync from your Peer you need to have the same state as you would sync from server. For keeping the state in ownCloud we use modification time and ETAGs. For more datails please go through articles in @dragotin blog https://dragotin.wordpress.com/2015/03/13/owncloud-etags-and-fileids/ and https://cds.cern.ch/record/1970463?ln=ru
Anyways, along with the technical details, for all the C++ code and client side related project we would encourage you to submit some patch during application and between 4-30 of May (introduction session with mentor, most probably me with support of other colleagues), we will later announce what patches that could be, so that you get good introduction to the code.
To give you an early feeling what is required for this project is definitely knowledge of C++, some understanding what is Qt (but not required), and relatively strong feeling on HTTP, networking principles. Good to know will be also what is WebDav. During the project you might also need to write some PHP code on the server side (security, keeping state and knowledge on connected clients) for your prototype to work and later be a production code eventually.
Maybe also this topic will give you some feeling how to implement sync optimizations in ownCloud https://central.owncloud.org/t/improving-data-sync-process/2517 -> This was my student prototype project to assess how much sense for us might have HTTP2 and Bundling (https://www.youtube.com/watch?v=9RecQUqddfs)