Weave isn’t limited to just syncing those types of data — cookies, history, tabs, etc. — which have support built-in. It has a client API which allows new sync engines to be written and plugged in. A sync engine implements some logic for reading, writing, and updating a data type; the Weave core handles the networking, encryption, and synchronization algorithm.
We’ve recently written up a Wiki page of Weave client API documentation, which explains how to write and install a new sync engine. We hope that having this API documented will inspire the community to experiment with extending Weave to synchronize new data types.