Honestly this thread looks like a fun place to discuss the mons we've used in the past in game while growing up, and I'm thankful for that. I was looking at the
poll results from a post from the previous page while writing this up.
If I were to name a few factors as to what I think makes a mon popular (from before getting into the competitive pokemon scene or in other words as as a kid that grew up playing the games):
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If it evolves (preferably 3 stages): The starters were fun to use because you could see them grow when they evolved and it was always fun evolving them and getting a powerful looking pokemon at the end. Evolving twice might have made kids think that stage 3 pokemon were better than stage 2 mons (like Pidgeot vs. Fearow) and led to a higher desire to use pokemon that evolved more. Ampharos and Luxray and Staraptor kind of fit this category too (ignoring the fact how good they are once they're fully evolved), but having 3 mons that evolve twice sorta feels like having 3 starters in a way, and it made it more fun to see them all evolve over time.
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Typing: Everyone likes having a fire, a water, a grass, a flying and sometimes an electric type too. You get your starter and then it's easy to want mons to compensate for the starters you couldn't choose, not to mention they do well as a trio in terms of coverage. Flying types are always early and generally just really helpful and for the most part they're pretty solid (Swellow and Staraptor are 2 that are hard to not pick up because they're great and they're available early and they get Wing Attack early as well).
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Evolution methods/Availability: Trading in particular is what I had in mind for this one, as I was planning on mentioning how a lot of people don't go with something like Golem as a rock type. Back in the day, it was harder to find a friend to trade with and that usually meant not being able to use mons that evolve by trading, and for stuff like Golem and Machamp, it made it harder to want to use them since they would never be fully evolved. It's easy to want to use mons that are fully evolved by something like 36, as you'll still be fighting gyms and have time to level up moves and have them to help out against gym leaders and bosses. The 2 exceptions for this come to mind with Alakazam and Gengar, as Abra you can get early in Red/Blue and Gastly is in Lavender town, and their final evolutions are really good and cool.
I know for a fact that in Ruby and Sapphire, Feebas was insanely hard to evolve since you had to max out its beauty using pokeblocks, and you could only give it a limited amount, which would result in something I had with a Feebas (which was already hard to find I think) that couldn't evolve. That and getting people to do non plot related stuff like contests/pokeblocks when it's not necessary might've discouraged some people from bothering with that, as getting pokeblocks specifically for beauty wasn't easy as well. I love Milotic, but before the prism scale, it was just really hard to obtain and something I'd never pick up. Friendship was also sorta tricky, but it was nice as well since most of the time it doesn't take forever if you're able to use that mon for a while and have it still put in work (something like Golbat and not Togetic).
The last part of this section is gonna be about the levels at which a mon evolves at. As mentioned before, something around 30-40 for the fully evolved mon is convenient if the mon is able to put in reasonable work until then. There's also cases where the final evolution is just really worth it like with Dragonite and Tyranitar (not to mention they're already popular to start out with) and it's cool having them. For pseudo legendaries at the least, the reward is worth the time, but for other mons (like some in BW), it's not the same with stuff like Rufflet evolving at 54 and Deino evolving at 50 and then 64 (which at that point is right around the E4 and champion level, and to not have a fully evolved mon by then is rough since you are most likely facing off against other fully evolved mons by then.
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Move list: This one is sorta self explanatory, and basically if you get good/decent stab options, then you're probably fine. This is one area the starters usually have covered well and makes them easier to use. In Diamond and Pearl (not Platinum where they expanded the pokedex) I think the only 2 fire types you could get were Infernape and Rapidash, and Ponyta was stuck with Ember and Fire Spin until level 38, at which it got Fire Blast. It got Flare Blitz later, but that was at 58 (not to mention it evolves at 40), which would be around the level you fight the Elite 4, at which point you really do want good moves by then. Infernape on the other hand at least had Flame Wheel for most of the game and had Close Combat, which made it a much more convenient option to pick. This also makes it easy to explain why Torterra works out, as a ground type that levels up Earthquake at level 32 is insanely good considering how other ground mons like Garchomp don't even get it until you get the TM for Earthquake, which is typically late in the game. This category also makes it easier to want to go with Typhlosion in Gen 2 or HGSS since it gets access to Flamethrower (and Lava Plume in Gen 4), both of which are good with it's decent special attack stat.
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If it was in the anime: This one might seem odd, but as a kid that watched the anime alongside playing the games, it's easy to say that I was motivated/geared towards using some of the mons from the anime that looked cool. Ash having a Charizard and a Butterfree might've had an impact (and might explain why so many more people go Butterfree over Beedrill), and another example would be Lucario. Lucario got his own movie and me personally I loved it and thought it was a really cool mon from the start and jumped at the chance to get one. Dragonite also had plenty of good moments to be treated as this almighty pokemon that always looked powerful. This category isn't super important, but it's more of a possible reason behind the popularity of some mons.
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Design: This one is hard to describe since everyone has a different preference for what they like, but if there's any evidence that it matters, it would be how Rowlett/Decidueye had 50% of people choosing it as a starter (compared to Incineroar and Primarina who both had ~30%), but it also ended up as the top mon in Alola. People loved Rowlett as soon as they saw it and Greninja is another example that ended up as the top mon in Gen 6.
I don't wanna make this too long, so I'll end with this:
If you look at the top pokemon by type for Sinnoh (which excludes the starters and doesn't count normal + flying types under normal), you'll see that pretty much every pokemon in the top category is a really good or strong mon, except for one. Idk how he didn't make the top 10, but everyone knows that in DPP, the real MVP of those games is none other than Bibarel. I'm glad to see that he and Linoone (for Gen 3) made the list since those 2 are the best HM slaves for those games, and they truly do help out with dealing with having to use HM's that we wouldn't want on our main party members.