National Security Projects

Agents left in your Headquarters help out with your National Security Project. This is a global bonus that modestly helps all of your Cities or agents. There are several options for these Projects, and they get the full bonus for every agent you leave in Headquarters. Therefore, you can stack several of them for a higher boost. Change the Project whenever you wish, but know that this causes a delay for several turns before you start getting the new bonus. During that in-between period, you don't receive any benefits from having people at your Headquarters.

National Security Projects

Homeland Security +1% City Strength and City Hit Points for each Agent at Headquarters

Research and Development +1% Production for Wonders for each Agent at Headquarters

Propaganda +2% Health for each Agent at Headquarters

Operations Support Lowers the risk to Covert Agents doing Operations for each Agent in Headquarters

Operations Reconnaissance Increases the chance of Operation success for each Agent in Headquarters

All of these bonuses are fairly minor, so it's not especially lucrative to have many of your Agents stay at their Headquarters. That said, there is no risk or downside to leaving people there (except for the opportunity cost from avoiding Covert Operations), and in the right circumstance, such as a high-risk Operation or an important Wonder rush, the Project can be very helpful.