We're currently using Lync via Office 365 but we want the Persistent Chat feature. I don't think Office 365 supports this -- is it possible to add this capability by installing our own Lync server and somehow connecting it to Office 365 for authentication? Is this what a Lync Hybrid Deployment installation is for?
This is an old question, but I'm answering it in case someone else comes across it:
It is still true, as of October 8, 2014, that Lync Online does not offer Persistent Chat, as per the service description. (cf. http://technet.microsoft.com/en-us/library/lync-online-service-description.aspx) There have been no announcements suggesting this is changing any time soon.
Hybrid can refer to a mix of Lync on-premises and Lync Online, or a mix of Lync on-premises and Exchange Online, etc. - there are a few possibilities.
The general use case for Hybrid Deployment is Enterprise Voice (cf. http://blogs.technet.com/b/nexthop/archive/2013/05/13/lync-hybrid-offers-simplified.aspx). As of today, Lync Online offers no PBX integration/replacement scenarios, so if you want full Enterprise Voice, you need to use on-premises. This includes PSTN (public switched telephone network) calling. However, Microsoft has announced this is changing, and somewhat soon (cf. http://blog.insidelync.com/2014/02/perspectives-on-the-big-announcements-at-lync-conference-2014/, http://www.nojitter.com/post/240167129/lync-online-enterprise-voice-out-pstn-calling-in) - if the part of Enterprise Voice you care about is PSTN calling, then Lync Online will cover it.
Persistent Chat could be used this way by having the users who should have access to that feature homed on-premises and users who don't homed in Lync Online. Or, you could just ignore the Lync Online portion of your Office 365 deployment. However, if the only feature you're going to use is Persistent Chat, and not Enterprise Voice, then this feels like a lot of effort to go through just to get that benefit. There's a lot of buzz around Slack lately as a hosted chat solution - that might be a reasonable way to go.