I'm kind of stuck on the idea of what a "secondary starter" really is, tbh. People have posited some good examples, but IMO there's a big difference between "this is a strong Pokemon available comparatively early-on" (of which there are many) and a Pokemon the game actively positions as a backup or alternative starter (of which there are few). There are quite literal examples of "secondary starters", most obviously in XY, in that they're Pokemon the game pushes into your arms and basically tells you to use. Outside of that nothing in the games explicitly tells you one Pokemon is more worthy of your time than any other (outside of instances where an NPC might say "I need a Grass-type Pokemon if I'm to have a hope of defeating [
gym leader]" or something.
Some thoughts off the top of my head for various titles:
Kanto
It's Eevee. Well, kind of. Even though it comes relatively late, in the RBY games Eevee is potentially something of a clutch mon as it's a cheap and easy way to add a strong Fire- or Electric-type to your team, both of which are fairly elusive and likely something a lot of players will be missing (the same can't really be said of Water, though at the point in the game at which Eevee is obtained there's still many Water-types the player won't yet have encountered). Despite being a 2-stage line it's comparable to first-stage starters in terms of stats, which is why it's been used as an actual starter in some titles.
Honourable mention goes to Pidgey for all the reasons
MegaFlareon noted: it's one of the few early-game rodent/bird lines to be a three-stage line and has a very similar evolutionary pattern to actual starter Pokemon. It's built to be a dependable option and it absolutely is.
Pikachu is in a funny mid-ground of being a Pokemon that's not quite in the same boat as the former two examples - it's a useful and powerful Pokemon found very early on, but one that's extremely easy to miss - that later
became an actual starter in Yellow. So it's not one that really counts, and in most of the other games it's found in comes way too late to ever be a viable secondary starter: in Johto it's post-Elite Four unless you hatch a Pichu from the Odd Egg in Crystal, in Hoenn it's in the Safari Zone, in Sinnoh it's solidly midgame, in Unova it's not there at all. Ironically of course Kanto makes its own actual starters secondary starters in both Yellow and LGPE, since you start with other Pokemon.
Johto
I'm not really convinced there is one for Johto. In large part this is because Johto shares so much with Kanto, so both Eevee and Pidgey reprise their roles from RBY - even being available in the same manner at pretty much exactly the same stages of both games.
I mean, yeah, sure, Mareep too. But it's not encountered quite as early on as a lot of other Pokemon (in the originals at least - HGSS made it possible to catch before the first badge) and it wasn't really pushed that hard in GSC; infamously, it's not even available in Crystal. It's certainly a crutch option in those games, but I don't really consider it a secondary starter to the same degree as Eevee or Pidgey.
Hoenn
As others have mentioned, Ralts is this to a certain extent in RSE. Despite being miserably hard to find on a single route, it features in an unskippable cutscene so it's one players actually know to look for. It's arguable how much that's RSE really pushing it at you though - I can imagine some people not wanting to use a Pokemon their rival is using, while others won't care about that and will want to use it anyway.
It's not exactly a Pokemon you'll fall back on in the earlygame though as it's incredibly weak compared to comparable Pokemon, but obviously it gets better as it evolves and definitely rewards the effort spent on it. Three-stage lines can often feel the most rewarding to raise up and it strikes a good balance in terms of when it evolves - 30 is relatively early for Hoenn, but it's just late enough that it won't dominate the game once it's fully-evolved.
And yet it's just a bit too niche for it to feel like a secondary starter really. The fact that it's so rare means a lot of players won't end up using it and it's weak enough that a good amount of people may have junked it out of frustration. Emerald even has an in-game trader requesting Ralts so I guarantee there were some players who traded theirs away.
In the remakes, Steven's Beldum, if you got the event distribution, is set up as the perfect secondary starter (it's even got Hold Back for extra in-game utility). I actually completely missed this distribution initially, but if I ever play through ORAS again I'm absolutely using it.
Sinnoh
I mean yeah, Starly and Shinx. I don't think there's much that hasn't been said about these two, but imo they're both very much in the "this is a strong Pokemon available comparatively early on" camp, not the "secondary starter" camp. But Sinnoh's roster is so limited there's not a lot else that really applies. It's interesting Budew isn't talked about much in the same vein as Starly and Shinx because it's another three-stage line available very early that's extremely good once fully evolved, and is even a Grass-type so is the natural option if you didn't pick Turtwig. I guess it's just less attractive in its base form since, as a baby Pokemon, it's not as immediately powerful as Starly and Shinx can be.
Unova
I mean this one's a gimme. The game literally could not do more to position these three as secondary starters if it tried. Cheren and Bianca both use one, the trio is used repeatedly in various instances throughout the games. It's just that, uh... not many people liked them. That doesn't change the fact that they very much are actual and literal secondary starter Pokemon, though. They even get Overgrow, Blaze, and Torrent as their Hidden Abilities for god's sake. Nothing else in Unova comes anywhere close to having a claim to "secondary starter" status than these.
Kalos
Not a Kalos Pokemon, obviously, but I don't think there's much of a better example than this one. The game does just about everything it can to force you to take it and use it. In a way that's very, very cringe, but that's a conversation for another day.
Speaking of cringe things the game forces on you, Kalos, of course, gives you an actual second starter Pokemon! I mean... does this really bear any further discussion? Not really. And it doesn't stop there: if you got XY right away you could also potentially get a third starter in the form of a Torchic distribution. Imagine actually having a starter trio composed of Froakie+Bulbasaur+Torchic or Chespin+Squirtle+Torchic. Wild stuff.
You could maybe make a case for Flabebe too, I guess. It's specifically pointed out to you as "hey, this Pokemon can be caught here! Do you have one? Why not?" That's obviously because it's a Fairy and the game wants you to try out the exciting new type that exists, but Flabebe is obviously a natural choice as another three-stage line with a fun gimmick in its different colours. Given Sylveon's introduction, it's interesting that XY doesn't give you an Eevee in a similar fashion to Lucario to try and push you towards that (it's available from an in-game trade, but this is luck-based and takes a few days to unlock).
Alola
There's a few things in the Alola games that could qualify, but Rockruff is the most interesting one since, prior to SM's release, there was an obscure reference in CoroCoro ("
similar to the starters, Rockruff hides a secret") which got misquoted and mistranslated a lot and caused some people to think that the species had some sort of shared trait with the starter trio; it's generally accepted now that this simply meant that all four species had their own, not necessarily identical, secret(s), with the starters having unique Z-moves and Rockruff having a unique evolution method.
Regardless, that got people interested in it and Rockruff is very much a Pokemon the game nudges you to use. It's popularly featured in artwork alongside the protagonists, the Pokedex literally states it to be well-suited to beginning trainers, there was a distribution when USUM released for an Own Tempo Rockruff with an elemental fang. Yeah, this one was definitely made with the intention of being a secondary starter alright.
Beyond that, Alola gives you access to Munchlax and Happiny
incredibly early compared to other games, with Munchlax even being distributed in the same fashion as Rockruff at SM's release. Were those not baby Pokemon, I would say they're more in the "this is a strong Pokemon available comparatively early on" camp, but the potential payoff of raising them up (and especially the fact that Munchlax comes with a Z-Crystal attached, which screams "use me") means there's a strong, strong incentive to use them so they just about edge into secondary starter territory IMO.
Orre
Including Orre too? Yeah. Because Colosseum actually does give you a pretty straightforward starter choice with the Johto trio. You start off with Espeon and Umbreon, of course, but very early on in Phenac City you're faced with a roadblock: three Cipher Peons, one red, one green, and one blue, bar the city's three exits and won't let you pass without a battle. Each has the middle stage of the starter that corresponds to their colour available to snag, but you only get to battle one of them and so can't snag the other starters until the end of the game*.
XD has a similar, but different scenario play out at a relatively early part of the game. The player is presented with six differently-coloured Peons, each with a Pokemon available to snag that roughly corresponds to the colour of their uniform (Houndour, Spheal, Seedot, Baltoy, Gulpin, and Mareep). However, unlike in Colosseum it's not a choice as you're at perfect liberty to catch all six of them. Since you get given the ability to evolve your starter Eevee almost straight away this gives you a great degree of flexibility in designing your team since you can pick the type(s) you're missing. The way these six Pokemon are made available to the player has always felt to me like a very deliberate facsimile of the starter choice.
*side note: impressively, they actually took steps to ensure you can't cheat this to get all three. If you snag one of the starters and deliberately lose the battle, both of the two peons you didn't fight will no longer have their starter on their team