Now that we are pretty sure it rains diamonds on Jupiter and Saturn, I believe that a traveling alien civilization would find that no ore is valuable in the sense that they are not rare. I considered maybe knowledge could become a commodity to be traded and sold. Can anyone either come up with something they could use of value, food would not be one of them. Or how knowledge could be harvested and sold, I am at a dead end for an idea.
We'll give you our next Dalai Lama, Steve Jobs, and Nikola Tesla, for your Bob Hope, Marina Abramović, and Garrison Keiller.
We also have one Einstein--willing to trade for your Johann Bach, Beethoven, or possibly Chef Gordon Ramsay.
There's endless topics within which humans specialize and of which they attain perfect knowledge, as do alien traders of all kinds. The desirable knowledge, whatever it may be, can be best attained through teaching from first hand experience by experts in those fields.
Pick any topic in the world and you will find its expert. You will find an alien interested in learning about it, too. And vice versa.
Some of the best fun happens when an entirely new topic is introduced to a world.
We do this today through a number of methods, and there are explorations of other systems. Setting up these methods may seem daunting efforts over decades, but the value of trading information is so mind-bogglingly huge that everyone wants to play.
The Patent System: by making an invention and promising its eventual entry into the common good of the public domain, the governments agree to grant a limited term monopoly over the use and sale.
The Rulers Reward: as in the novel Interference, the ruler or a representative listens to the merits of an invention, pays a one time cash reward, and places a record of the invention in the public domain. The rewards are large, but the value of the information is additive to soooo many people.
The Scarcity Method: as in Signal Shattered, the trick to making information valuable is to keep it from spreading. Find a race with a new trick; learn all you can and apply it to your industrial base; kill the race to keep a trade secret. Kind of works if you are big enough.
The Blackmail Method: not all information is of positive value. The Zorgzra require the following number of sex slaves or they will broadcast in the clear to all your citizens how to make antimatter bombs using just a few common kitchen implements.
It might be fun to have competing groups using different methods.
Short answer results of evolution and maybe some cultural stuff.
Life on diffrent world is going to be very different, we don't know how much chemistry we will even share in common, but we do know many biological products will be unique to their worlds.
If we share some basal genetics, unlikely I admit, then all of earths genetic diversity is valuable. but even if we don't there will still be a great deal of interest in research.
Food, this depends a lot on biology if we share some basic macromolecules then foods would tradable and many would be unique to each world, milk and milk cheese are almost certainly be unique to earth. If the macromolecules are different you could even have the dieters dream, food that has pleasant tastes but due to differences in metabolic pathways have little to no caloric value. The sky is the limit depending on how different you make them, "Oh you have to try these cow teeth and birch bark I got from earth they are amazing, so crunchy and delicious".
Flavorings and drugs, Larry Niven's draco tavern messes with the idea that one trader makes a fortune off unique flavors of liquors from different worlds. Each world has different flavors becasue they start with different things, Earth has Gin and Beer but maybe X has something that has a flavor we can't even describe becasue it is too different. Spices would be another option as would drugs, maybe riboflavin is their nicotine or their capsaicin.
Other biological products, anything biological has a chance to be unique to one planet. Earth is likely be the only world with feathers for instance, they are just too unusual of a pathway to happen a second time and there are other things that may not evolve on other worlds. Silk or wood could be unique to earth, both might be high demand products. Maybe the aliens have a life form that makes something like foam leather or natural plastic that is valuable to us but ubiquitous to them, maybe earth plants make the the best paper or really cheap ablative reentry plating(not as funny as it sounds, oak paneling is amazing reentry plating, it is just only good for one use) maybe all their animals have scales so earth leather is better for certain things or maybe strawberries are the perfect source for one of their expensive hard to synthesize cosmetics.
Pets and the like, maybe they just think rabbits are awesome or really think crickets are amazing. Exotic plants and animals have always had a market.
Novelties, humans certainly like to steal cultural artifacts from each other why not aliens, then you have cultural fads. Maybe hair is in and humans make the best wigs or the tridions really like impressionistic art, or disco, or human coins.
Most of these answers are found in the questions mentioned as possible duplicates, but I think that the following discussion still has some value.
First of all, trade requires different goods that can be exchanged. So instead of asking "What good would they trade?", the question should be "What goods or services would they bring, and what goods or services would we give to them in exchange?"
The simplest commodity for this situation is Energy. To quote the Hitchhikers Guide to the Galaxy, "Space is big, really big..." and to get from one place to another in a reasonable amount of time requires a lot of energy. If visitors were to come to our solar system with certain things we want (John gives some good examples), they would come asking for energy. If they were able to load up with materials like hydrogen and uranium when they visited us, it would make a good stop, especially if they didn't have to mine those things themselves. The reason they would prefer to trade as opposed to mining themselves is that mining requires equipment, which means they have to bring that equipment, making space travel much more expensive.
There is also another advantage that Energy has; namely the fact that it is both exclusive and rivalrous. This means that it can be treated as private property and that when one person consumes it, others cannot. These are almost essential to using them as trade. The issue with using information as a trading good is that it is neither of these. There is no way to control it as private property (without some higher governmental enforcement agency), and the fact that you had an idea does not keep me from having an identical idea on my own. Wikipedia has a good discussion of the topic, and you might be interested in doing some further research on the topic.
As a result it might not be that people trade knowledge per se, but instead basic resources (energy and materials to repair ships) and technology, or implementations of ideas. Maybe everyone knows that splitting water makes two hydrogen atoms and an oxygen, but if one species has figured out a way to split them with as little energy as possible, that is technology. That is something people might trade for.
In summary I would suggest that you study the following economic principles: Exclusivity, Rivalry, the economic concept of Utility, Opportunity Cost, and Comparative Advantage. These are the fundamental principles that modern international trade is based on, and intergalactic trade is just a much bigger version of the same set of problems and solutions.