Building a Stronghold:
the cheapest of which is either a trading post or guildhall, costing 5000 to construct and 60 days downtime to complete. The highest is a cool half-million gp castle taking years.
Carousing: basically bar-hopping, going to parties, gambling, binge drinking, doing drugs, etc. Results vary including gold rewards, new contacts, jail time, romance, etc. XGtE has updated rules from the DMG I prefer.
Crime: XGtE introduced this as somewhat an expansion on Carousing. It is a 3 check skill challenge. The number of successful checks determines what happens. Best case scenario is 1000 gp of loot. Worst is jailtime for a number of days.
Gambling: another extension from the DMG introduced with XGtE. Like Crime, its a 3 check skill challenge. The payout depends on your ante, with the best case doubling your bet.
Pit Fighting: A 3 check skill challenge paying out up to 200 gp.
Relaxation: if you have a long term ailment requiring daily/weekly saving throws, taking this Downtime activity gives you advantage on the saves to recover. You can also use this to end an effect lowering your max hp, and even to recover lost ability points (such as a reduction in strength or intelligence). This works for poisons and diseases, but not most spells with long durations.
Crafting a Magic Item: including rules for prices and rarities. This includes spell scrolls. XGtE has updated rules from the DMG I prefer.
Gaining Renown: I don't really track this optional statistic, but have used something similar in a different game with more complicated faction interplay. Basically it leads to stuff I give you anyways like adventure leads, safe houses, new traders and could represent your 'rank'.
Performing Sacred Rites: weddings, funerals, ordinations and such. The benefit is that you get Inspiration every day for the next 2d6 days.
Religious Service: XGtE has an alternative to Sacred Rites, where instead of earning Inspiration you get a few favors from NPCs. You can 'pool' favors for later use, and you get a pool max equal to 1+cha mod.
Research: learn lore depending on an intelligence check. The check can be modified by spending gold on research costs.
Running a Business: Pretty much a random roll to see if you make a profit or lose in maintenance costs.
Selling/Buying Magic Items: pretty self explanatory. XGtE has updated rules from the DMG I prefer.
Sowing Rumors: swaying public opinion. DC 15 Deception or Persuasion will effect a population for a certain (rolled) amount of time based on the population size.
Training to Gain Levels: I don't require this optional rule, but you could still RP developments as you like to explain new powers.
Training: The original language/tool learning rules were updated in XGtE. However your character's intelligence mod now reduces the time required (but not the gp costs). For example, with 18 int the time required to learn Infernal would drop from 10 workweeks to 6.
Work: a single check determines pay. Checks that include tools have the best chance to reach a high DC and get the highest payout (basically covering your lifestyle expenses along with a bit of pocket change).
Creating your Own Downtime Activity: I'm pretty open to whatever you'd like to do that isn't on this list.