I started writing this article before our first official challenge, so now when things are calming down I finally got to finish it!
Synergy is a card effect where a card grants a synergy bonus to the other cards in the deck that has a trait listed on the card. Keep in mind that the synergy bonus applies to the Quality Level of the card, which will increase ALL skills of the relevant character or ship, thus providing a very powerful and scalable bonus.
Before diving into the article I want you to think about: "What does a Weapon Officer has in common with Captain Cranium?" (beside both being Space Combat Officers). An answer to the question follows later in the article.

Synergy overview

I have used the deck expert tool to look at all the current cards granting synergy. I have not included cards that we know exists, but have not yet been revealed (e.g. corporate synergy bonus for Mysterious corporate boss). I have not looked at the similar but different bonus of "inspire" cards, this topic may warrant its own article.
Currently there are 12 cards granting synergy bonus, of which 7 are origins, 3 are spaceships, 1 officer and 1 citizen. Keep in mind that origins are mutually exclusive - you can only pick one per deck - and so are spaceships, so currently in practice it is not possible to pick more than 3 synergy granting cards in the same deck.

Number of traits affected by synergy

As you can see in the table above, there are 18 traits that can be affected by a synergy bonus currently. Five origins provide a synergy bonus to 3 traits, one to 2 and one to 1. Only 4 traits can receive a +2 synergy bonus - Spaceships, Science, Criminal and Nomad Fleet. I.e. a Navy Lieutenant with a Quantum Class Supreme receives a +2 spaceships bonus, an Ark Scientist with a Coetus Science Vessel gets a +2 science bonus, a Repentant Pirate with Captain Cranium has +2 criminal and a Taurus Transport gives a +2 bonus to Nomad Fleet.

Synergy to self, background

Currently there is one officer card with synergy bonus (Captain Cranium) and it provides a synergy to itself for its criminal trait. It also provide another synergy point for its background, since it always receives the "criminal" background, which in turn grants a bonus trait "criminal". This seems to trigger another synergy, for a total of +5 with the Repentant Pirate + Cranium combo.
My personal opinion is that this is not a consistent behavior with other cards and suspect it will have to be changed, but it can be that this behavior stays. I will anyway raise it for further discussions in discord at some point. My largest issue is currently that the trait "criminal", which I would consider relative common, has a corresponding background "criminal" that provide a bonus trait "criminal". The trait "scientific", "military" or "civilian" does not have any background that grants the same bonus trait. It can of course be that Elindos has future plans here that will provide more balance between the backgrounds and faction traits. The other aspect I find unbalanced is that if you have a faction trait "criminal" and receive a bonus trait "criminal" you receive another synergy bonus. It seems that this double bonus is only generated for the origin card currently, and not the Cranium officer card (otherwise Cranium would have received a +6 bonus when used with Repentant Pirate).
Another point to think about is whether origins are providing a synergy bonus to themselves, which is not possible to assess at this point in time, since the QL of the origin has no impact on current gameplay, and is also not visible in the deck builder or launcher. I will go back and evaluate this later on, when we know more about effects of origin cards QL.

Missing ship synergies

The synergies provided by the Quantum and the Coetus ships currently does not work, but has been reported as bugs, so I expect them to be fixed soon. The Taurus ship synergy is difficult to test currently, given the lack of Nomad cards, but I suspect that synergy on this ship is missing too.

Cards receiving synergy

In total 69 cards has traits that are relevant for the existing synergy bonuses, and of these 13 are ships. Increasing the QL of your ship will make it stronger and being able to sustain more damage, beside other effects that is still to be revealed. 18 cards are officer cards, 34 are crew and 4 are citizens. I have not included any passenger cards that are generated at the time of colonization (such as science student), since it is not clear yet if the synergy bonus will apply or not.
Below I have compiled a list of how many cards have a trait that activates a synergy, thus giving an idea which synergy bonuses have the widest reach in the full card collection.

Given the list above, we can clearly deduct that the Navy Lieutenant has the most recipient cards for synergies, with Military and Warfare being the 2 most common traits among the synergy traits, and spaceships being position 6.

Synergy score

In the list below, I have scored all the synergy granting cards based on the sum of all cards having the trait for each of its synergy. There may be some double counting here (for example many military cards also have the warfare trait) but it should still be a good proxy score since double counting receive +1 synergy for each trait that matches.

Based on the list above you can see that the Navy Lieutenant clearly comes out on top with a score of 50 versus the second at 29. I was actually surprised that the secret agent came out second, it has not been on my radar before as a strong synergy card. The Kumicho is scoring deceptively low at 10, given that it has a secret weapon: Any card that is equipped with either of Syndicate chip or Narco-Warfare will receive its bonus, providing a powerful bonus! The Narco-Warfare is to my knowledge not working yet, so we may need to wait until fixed before we see this synergy bonus.

Summary and answer to the question

In this article I tried to provide a closer look at the synergy mechanics as currently implemented in Exode. I suspect we will see several expansions soon, the mysterious dark boss for example is rumored to provide a synergy bonus to corporate, and I expect the legendary Mysterious AI card to do something similar to Robots.

The Navy Lieutenant came out as a strong starter card for synergy and that actually brings me to the answer to the question I asked: A Weapons Officer receives a full +3 synergy bonus from a Navy Lieutenant Origin, and so does Captain Cranium from the Repentant Pirate. These were the only 2 cards I could find that received a full +3 synergy bonus from their origin cards (please let me know in the comments below if I missed any cards here). So save your Weapons Officers, they may get some extra value for their strong synergy - not bad for a starter card!