I prefer everything that the game offers in a unique way, and that you can't get otherwise. Or not easily.
For example, if I am offered a choice that gives more money, I am not interested. Because if the situation becomes problematic, I can just build money quarters. I can already get that element of the game in a normal way.
For stability, it's more difficult, but you can get it.
For influence, it's definitely difficult. The only way to get more influence is to have a culture that produces a lot of it. Otherwise, buildings or infrastructures that give influence give little, or are difficult to get in the technology tree. So influence is a good choice.

Some of my favorites:

Wonder: Lighthouse of Alexandria.
+2 Movement bonus for naval units. Hard to get other than very occasionally. If a tenet or wonder allows you to get it, I'll take it. This makes the lighthouse of Alexandria one of my favorite wonders. Moving through the water slowly is a pain. Especially for land units tasked with exploring or conquering an island.

Politic: Small Council.
+1 city cap. The city limit is hard to change, there are some techs and culture for that, but otherwise, nothing. While the other choice is 25 stability, but only on the capital.

Politic : Republic.
+3 strength on unit. It's possible to get strength bonuses in various ways in the game. But not that much. Except for the infrastructure which increases experience. So a raw bonus on units is better than having +5 money on a main square.

Politic: Plundered Wages.
+50% money on ransack, +4 force when ransack.
Excellent if you are militarily dominant, you have a lot of money. However, at some point in the future, you will need to revise this policy and then choose Paid Wages.
Paid Wages: +20 Stability Stability on City that you occupy. +5 Stability Stability on Garrison.
Every stability bonus on garrison is important. Garrisons are not very popular with players for stability. But it also has the advantage to make armies pop where you want and to boost them. But above all, if you accumulate the various bonuses on garrison, you can reach +35 stability. This becomes very useful if you play Pollution without having the means to lower it yet, and if you are going Soviet in the last era. Which I often do.

Politic: Open-Minded.
+3 influence. I prefer this to having +5 faith, even if at the beginning of the game it is interesting too. But I struggle more with influence. In the long run, it saves me from revising this policy too. Because I wouldn't use faith anymore at the end.

Politic: Freedom of Speech.
- 20% costs of politics. I prefer this to something that prevents a revolution, because I make sure I don't have stability problems and avoid revolutions.

Politic: Monoculturalism.
+5 stability on city or administrative center. I prefer this to +1 influence.

Politic: Inherited Land.
Produces outposts and ties them with money rather than influence. This policy is broken. Just huge. No more influence problems. No need to waste a culture for influence. Build more money quarters, especially on resources and it's unlimited expansion with easy management of your empire (remove an administrative center from one city, give it to another). This also works for absorbing cities.
You can produce your cities normally at the beginning, then build a megalopolis of 24 provinces. You can even raze your own useless cities (by transferring their pop first). If you haven't tried this yet, do so.
The other choice is more obvious "Communal Land", +10 food per territory attached to a city. It seems powerful, and of immediate use, but it is a trap. I much prefer the other choice first. Eventually, it is possible to switch from one to the other. For example, choose "Communal Land" once your continent is in place or you have excess influence.

Politic: Sixteen Resources.
+30 war support. I always choose this because the other choice offers me to lower the number of deposits to make a luxury extractor. But I always have enough, or I conquer it. So I prefer the war support, which can only be obtained marginally. But both are interesting.

Politic: Public Security.
- 50% garrison cost. I prefer that to -50% cost of the commons district. I hate the commons district. Organizing the layout of the quarters is complicated enough with the synergies. I don't need to be burdened with that on top of it. The commons district screws up the synergies on top of that. They are hard to place (where there is a neighborhood transition). I used them to get the small influence boost rather than the stability.
When you play garrison, you need to spam this fast. You can even conquer a big crappy island (no resources), only to place a garrison forest there if you need stability. No need to complicate your life with commons district placements, which you will always build in limited numbers anyway.

Politic: War Slaves.
+1 Population Bonus per Ransack.
Another broken policy! Very strong, if you are militarily dominant, and have one or two favorite victims nearby. Loot a resource depot. Wait next to it for a turn or two, the AI rebuilds it, loot it again!
The other choice is a bonus on the commons district, which you'll never build anyway.
Looting is so effective that you forget to conquer at least one city and lose the war. Don't forget to win the war after looting a lot. And don't bother to take the cities, take the money. Manage to declare war soon, and loot again!

Culture: Achaemenid Persians.
Classical era. In some circumstances this culture is inevitable. It is your only choice, your savior!
Just for one thing: +2 city cap. Right now! When it's the only thing you need. With that, a building that generates influence, not a huge amount, but what you need at this stage of the game. Very useful.
You make this choice if you have a convincing military start. You can conquer another civilization's city (and wipe it out at the beginning of the game). You take other cities from free peoples (peaceful or hostile). And at the end of the ancient era, you end up with 6/3 cities. Such a huge influence deficit (a debt that will add -200 of stability per turn) that you will be forced to release the cities. The Persians save you from this. You start the game with two to three times as many cities as your opponents. It's a total snowball.

Tenets.
Often, I get outplayed by AI opponents, and I take the tenets they leave me. However, I am quite happy with some of the choices. There is often a tenet or two left over from the military. And they turn out to be more useful.
Of course, if I have rivers, I would like to have +2 Stability on River. But if it's already taken and I have Smite Unbelievers, or Hunt the Infidels left, I can live with that. They offer hard to get bonuses, while Stability you have more ways to get.

Tenet: Protect the Weak.
+5 stability on garrison. If you have accumulated another bonus on garrison, it is valid to press it. Otherwise, forget it. It is better to Undertake Pilgrimage, which reduces by -25% the political costs (obtaining as well as cancelling), which climb very quickly in cost.

Tenet: Eschew Gluttony.
The only choice I like. Even if they let me have the rest, I'll take this.
+1 Movement Land Movement Speed on Unit.
+1 Movement Naval Movement Speed on Unit.
-10% on Unit Industry Industry cost.
The bonus in movement speed is priceless.

Tenet: Meditate Often.
+2 strength on Units. Even though I have built many cultural wonders, I hesitate to take Raise Monuments (+50 science on wonder) rather than Meditate Often. It depends on your game balance. If you have a lot of bonus % to science and a lot of wonders, then it is better to take Raise Monuments. If you don't like to suffer military defeats and win battles even in low numbers, Meditate Often (combined with the other bonuses chosen before).