These notes were taken mostly from a post from Caith during beta, and are pretty much all we have with regards to an explanation of how the system works. I have added in comments where appropriate, on things that we know changed between beta and launch. I also put together a quick video regarding the Overseer system.
As of the beta update for 11/23/2019 the Overseer system is now available on Beta. To gain overseer quests and agents purchase the An Overseer's First Agents crate from the beta buffer Finch. This will grant you the short tutorial overseer quests and your first agent. Completing the tutorial overseer quests will grant you another agent, as well as your first two persistent overseer quests. For the purposes of beta, the overseer rewards only drop more overseer system agents and quests.
Mum note: That was the beta-only way of accessing your overseers. For live, you are looking for a Overseer Starter Pack in the Promotions dropdown of your /claim window, NOT your BoL dropdown. That starter pack gives you your first overseer and the intro overseer quest. That quest completion, in turn, gives you another overseer and a second intro quest. That second quest gives you your first regular overseer quests.
How do I open the Overseer window? The Overseer window is listed under Overseer in the main menu.
Mum note: "Main menu" means the EQII button.
What is an Agent? An agent is a character you have formed a contract with to perform tasks throughout Norrath. While all agents can help fulfil an overseer quest, higher tier agents have specific traits which may positively or negatively affect the outcome of an overseer quest. In game systems terms you collect these agents as in game items and add them to your overseer agent collection, and assign them to overseer quests. Once used on an overseer quest an agent goes into a cooldown period to recuperate.
What is an overseer quest? An overseer quest is a task that you send agents to accomplish. Each quest has a set of positive and negative traits that can affect the outcome of the quest. Sending a sly agent may help in an infiltration quest, while sending an honest agent may well hinder that same quest, increasing the chance for a mishap. These tasks take a set time to complete and will always grant a reward upon completion with a chance for a bonus reward based off of the difficulty of the quest, as well as the matching traits of the agents you send.
Bonus Chance? A bonus chance is the chance the completion of the quest will grant an extra reward. You can increase your bonus chance by assigning agents that have positive traits for the quest, as well as assigning a mercenary.
Mum note: While you have no bonus chance for the two tutorial quests, the rest of the quests will have a percentage showing for a bonus chance. Use an agent that matches one (or more) icons for traits needed on the quest, and your bonus chance goes up. Click on the little shield icon directly attached to the top of the bonus chance listing to assign a Hire Anywhere mercenary to the quest to increase your chance at a bonus crate.
Mishaps? Every task comes with some risk, and each overseer quest has a chance for a mishap. When a mishap occurs one or more of the agents you sent on the quest will go into an extended cooldown. Assigning an agent that has a trait that the quest treats as a negative will increase the mishap chance on that adventure.
Mum note: You will receive a message in your chat window when completing a mission, if an agent had a mishap. The agent in question will have their cooldown time listed in red, and it will be over a day in duration. (More below about how to reduce that cooldown.)
Can I do anything about a mishap? First, avoid using agents that have a negative trait on the quest, though sometimes the positives may outweigh the negatives. The second thing you can do is assign a familiar to assist your agents on the quest. Assigning a familiar reduces the mishap chance, with higher tier familiars reducing mishap chances by a greater amount.
Mum note: Click the little shield icon above the mishap chance percentage to add a familiar to the mission to reduce the mishap chance. It will display available familiars with the highest-rarity ones at the top of the list. (The higher the rarity, the more it can reduce the mishap chance.)
What happens when my agent has a mishap? When your agent has a mishap it has either been injured or captured by another agency in Norrath. When you have an agent in such dire straits special overseer quests called rescue quests become available, which you can assign the unfortunate agent to decrease the extended cooldown time. Rescue quests, as with all quests, come with their own chance for a bonus reward. While painful, sometimes failure can lead to great things.
Mum note: When you start a rescue mission, the agent who had the mishap goes over in the lefthand box in the mission listing. Then you assign 2 other agents to perform the rescue, making sure to check for traits that will increase your bonus chance while reducing the chance at another mishap.
How many overseer quests can I run? Each account can start 10 persistent overseer quests each day. Cancelling an overseer quest will not restore the use of that daily charge. Some overseer quests do not go into this daily pool, as they are charged quests. An example of a charged quest would be the Become an Overseer tutorial quests. Once the charges on these quests are exhausted they remove themselves from your quest pool. Cancelling a charged quest will not restore the item that granted the quest.
Can I still use my Familiar/Mercenary while it is assisting with an Overseer quest? Your mercenaries and familiars may still be used while they are also assisting an overseer quest. Norrath is a magical place.
Mum notes: Players have noticed that they only have a certain number of their known overseer quests display as available each day. (In most cases, that number is 6, and does not include rescue quests in the count. Bunji had the following to say on the forums:
There are a set amount of quests from your uncharged pool (can do them more than x amount of times) that are randomly pulled and sent down to the client each day (shared among the account). So, you won't have all your unlocked quests available each day. When you first add a quest to your pool by using an item, etc, it is immediately available that day, but will then cycle normally the following day.
Quests for mishaps and quests with charges are handled differently.