1) Let there be lore, such as say based on extensive astronomical studies from Sol of Sag A, clouds, spectral analyses, M31 (Andromeda, visible in the skybox), and so on, that allows the establishment by Universal Cartographics of a standardised galactic plane and a 3D cartesian grid for reference purposes. The plane and coordinate system already exists, of course, it's just a lore thing at this stage based on things astronomers already do IRL to determine the galactic plane etc.
2) Let the positions of inhabited systems with populations over 1 million be known accurately and available in the Univesal Cartographics public database, using the grid from Point 1. Let the start-locations of lesser populated systems sometimes be similarly known, based on factors yet to be determined as gameplay, lore, or other factors, needs them to be, which other factors must include Point 3 overriding any gameplay or lore needs. Whether the internal structure of a given star-system is known in the public database is a separate matter, and can be left to gameplay and lore.
3) Let the positions of star systems within appropriate distances any stars from Point 2 above also be known, with the appropriate distances for knowledge of the location of a given star being determined by that star's luminosity. Naturally, various IRL high-profile stars would also be fully located, eg Sagittarius A*, Betelgeuse, Rigel, Polaris, et al.
4) There is no need for a new module, we already have our sensors, and we already have the FSS technique in-game. Let sensors be able to uniquely identify stars by their spectra, even if their positions are not fully identified yet. The better the sensors, and the more luminous the star, the further out it can be identified when seen in the skybox, where either it is a new star to add to our list of known stars or a star we recognise as having been seen before on our list. This sensors mode borrowing FSS techniques is for recognition and angle-measurement only, so it can be safely considered as built into Sensors and not require a discovery scanner or whatever else would be used for determination of structures of solar systems in a sequel to Elite Dangerous.
5) As we fly from one star to another, we take measurements of the angles of not-fully-located stars as compared to known referents in the grid. The more angles we determine by jumping to different stars and then taking further measurements against established referents, the more accurately we can determine the location of a star of interesty. I am aware that knowing the star class and apparent magnitude already affords an initial estimation of distance, but for gameplay at least I'd say it is not to enough accuracy to admit of safe jumping, therefore parallax-method and triangulation is the order of the day.
6) For jumping, let there be two levels of accuracy of interest. One is the minimum accuracy required to jump with some minimum confidence of arriving safe and within some reasonable distance of the primary star, but where there will still be a chance of misjump, and the other end is when enough measurements have been taken for the game to say in effect "Yep, accurate enough, misjumps will not happen, arrive not far from the primary star's EZ and just outside scooping distance." Commanders make their own decisions about when they want to risk the jump, so long as some mininum confidence level has been calculated by the Sensors. This minimum required confidence level could also be adjusted by Commanders, when docked, and perhaps only at an appropriate Engineer's Workshop?
7) When an explorer returns to civilisation and hands in data to Universal Cartographics, that data gets added to the public database that EVERYONE can see in their Galaxy Map if they have docked after the explorer has handed that data in (ie if explorer A is still out in the deep when explorer B hands in data, explorer A will not have an updated Galaxy Map showing B's findings until A's own next docking at a station with a Universal Cartographics service). Therefore, over time, the galactic fog would be lifted for everyone by explorers doing work bit by bit, either by playing alone or on organised mapping expeditions. Again, effectively, just bring EDSM in game. Think of it as one colossal and never-ending CG for explorers and with some universal rewards for everyone, if you will.
8) The above mechanics are for primary stars only, and not the other constituents of a given solar system - this latter has its own mechanics, which I wont go into here. As to monetary rewards, I'd keep the idea of first-discovered (and first mapped) bonusses for all bodies (primary star or otherwise), though, given the amount of time and effort required as detailed above, I'd jack up the firsts' rewards for all bodies considerably. Thereafter, a body only provides exploration monetary reward from UC for the next yay many Commanders to hand in additional data, say 1000? I see no immersive reason where there cannot be tiers of payouts based on how many times a scan / mapping of a given body has already been handed in, where actual reasons for not having something like that would be external to the game and be things like client computing power or internet etc. After 1000 (or other number) handings-in of cartographic data on a given body, UC will no longer pay out at all for more scans of that body. This allows for:
a) seasoned explorers to go out and do their thing on their own or as part of great expeditions, getting rewards for cutting new trails and further establishing new ones in a reasonable manner, without putting them all unduly at odds with each other, and
b) new explorers to start on their own trails close to civlisation and still get rewarded by starting where some people have already gone before and getting the itch to explorer and contribute more #ForScience.