If you request a mission from an agent and then decline the mission, he will not offer a new one. This is a bug (or glitch or oversight from the Devs).
I’ve submitted a bug report about this a couple months ago.
After you have started research with an agent, 23 hours later they will send a notification that they have “run into a problem”. You can ignore the message and you will still accumulate Research Points.
If you do the mission, you will earn a “days” worth of extra RPs.
One day the mission will be a short courier run with a tiny package (0.1 m3) to another station (which could be in the same system or the next system over or maybe 8 jumps away, in hisec or lowsec).
The next day the mission will be to “acquire” a small quantity of Tritanium. The amount depends on the level of the agent. A level 1 agent will request 300 trit, a level 2 will request 900, a level 3 will want 2,700 and a level 4 agent will want 8,100. (pro tip - buy a whack of trit and stockpile it in the station where the agent is. It usually will be cheaper than buying it from the market and saves you the hassle of having to transport more every time you visit the agent).
When you click “Complete” on the mission, the appropriate amount of isk will be deducted from either your cargohold or your hangar.
Doing those missions will also increase the amount of RP you earn per day by about 0.03 per day on average. That is the only way to increase your RP/day once you have maxed out your skills and have level 4 R&D agents.
If you decline the mission the only way to “reset” the agent so that they’ll send you a new mission notification is to buy all the datacores you can from him, cancel the research and then start it up all over again (losing any left over RPs you may have had).
I tested this as awhile ago as I have 5-6 agents on 15 toons/alts and sometimes the agent would offer a “courier” mission that would take you 4-5 jumps into low sec or 8 jumps in hi sec and it just wasn’t worth doing so I’d decline the mission. I noticed that the next day, the agent never offered me a new one. Waiting another day and still no new mission.
Finally gave up and bought as many datacores as I could, then cancelled the research, started it again and the next day was offered a new mission.
After it happened a couple of times on different toons I was finally able to track it down to being about declining the mission. CCP has acknowledged the issue and will fix it sometime in the future.