How to organize repository of common research papers for lab members?

The simplest solution to this problem that I can think of is just to use Mendeley (or service with similar functionality) that allows people to form groups (public or private), based on their research interests. Then, all activity, such as adding relevant papers, etc., automatically appears on group members' news "wall", plus they can set up e-mail notifications and such. For more details, see corresponding Mendeley's pages: https://www.mendeley.com/features/collaborate and https://www.mendeley.com/groups.

Obviously, solutions for more complicated research workflows will require more specialized software tools: open source, commercial off-the-shelf (COTS) or custom built.

Your problem is connected with creating of the Institutional_repository.

An institutional repository is an archive for collecting, preserving, and disseminating digital copies of the intellectual output of an institution, particularly a research institution.

DSpace is a software which helps you to organize an institutional repository.