Pauper is an Eternal format, like Legacy, drawing on 25 years of Magic innovation. Though new cards are added every year, either through downgrades or new printings, cards rarely leave. Cards don't rotate like they do in Standard, so the only way a card leaves the realm of playability is if the metagame changes to disadvantage a certain deck or if an individual card gets replaced by a better option.
Also, only ten cards have ever been banished to the Pauper banned list, and no card that has ever been banned in Pauper has ever been unbanned.
Because of these two realities, Pauper has a number of consistent strategies. Decks may change but archetypes rarely do. You can go under with an aggro deck like Stompy, over with a big mana control deck like Tron, sideways with a combo deck like Tireless Tribe, or play midrange like we see in today's Monarch Decks. Or you can just play Delver of Secrets and Preordain. That's always been a viable deck too.
Even when cards are banned, the archetypes remain: they just evolve. Cloud of Faeries becomes Faerie Miscreant in Delver decks; Cloudpost becomes Tron. The only archetype that WOTC has persistently labored to ban out of Pauper completely were the Storm decks since the strategy has few consistent answers in an all-commons format.
But if the card pool and archetypes remain the same, even when decks do not. What this gives to us is a consistent toolbox of counter-strategy cards. So when you're building a new deck you can always reliably consult on this toolbox as potentially having the answers cards you need. In fact, one of the truisms of Pauper is that variant archetypes, strategies and decks often share from this same toolbox of sideboard cards.
So what are these Magic bullets? (Pun intended)
Pauper's Sideboard Toolbox
Ray of Revelation is a pinpoint answer to problem enchantments, whether auras, Circles of Protection or opposing Journey to Nowhere. Other White anti-enchantment options include Leave No Trace, Kor Sanctifiers, and Patrician's Scorn.
Sunlance is that rare White direct damage and the cheapest removal option in White.
Circles of Protection are good control options, especially for Tron. The most popular colors are Green, Red and Blue. Don't overlook the Runes of Protection too, in Mono-White decks, since they also cycle!
Hydroblast is the most played sideboard card in Pauper since it excels in Izzet Delver mirrors, and against Boros Monarch, Red Deck Wins, or Burn.
Dispel is an excellent choice in Blue mirrors to win those counter wars or to stop removal.
Negate is the Blue catch-all. Rarely is it the optimal choice but it has a lot of utility in an open metagame.
Repeal is another very flexible card that benefits from Pauper's low average Converted Mana Cost.
Coral Net is as close as Blue gets to hard removal. It should nearly always trade for a card against aggressive Red and Green decks.
Deep Analysis is a great way to go bigger against another control deck.
Duress is Black's quintessential sideboard staple and is as great here as it is in any format.
Crypt Incursion is both a graveyard hate card and also a way for a control deck to gain an exorbitant amount of life, effectively putting the game out of reach for many aggressive decks.
Pestilence can be a recurring sweeper if your bodies are bigger than theirs or have Protection from Black, and it doubles as a finisher.
Wrench Mind is a very cheap two-for-one but suffers that many control decks, including Boros Monarch and Tron run many artifacts.
Pyroblast is the most flexible sideboard card in Pauper. In a Blue dominated format, I will run the full four in any deck that plays Mountains.
Electrickery is the most useful sweeper in Pauper since so many of the threats are X/1s. It's cheap, an instant, and versatile enough to play for in a pinch against Delver.
Gorilla Shaman is the single most powerful hate card in all of Pauper. I suspect that it is the sole card keeping Affinity from harsher bans. But other Red decks will prefer the flashback Ancient Grudge, which can hit artifact creatures, Prophetic Prism, or artifact lands, or they may prefer Smash to Smithereens, a great racing option for Burn.
Stone Rain is the most basic flexible land kill spell Red can offer. But specialists might prefer Earth Rift for its flashback, Raze for its cost efficiency, or the double Red Molten Rain for its 2 damage rider.
Swirling Sandstorm is an incredibly powerful but specialized sweeper. Until Izzet Delver, a deck that runs 12 out of its 19 creatures as flying threats, no deck had ever been able to run it effectively. But in that deck, it can wipe out all your opponent's ground threats while leaving all your fliers untouched, provided you can reach Threshold.
Flaring Pain is one of a kind in Pauper that it can negate damage prevention.
Natural State is the cheapest Naturalize around and only "misses" on a few of the largest CMC artifact creatures. Nature's Claim is also an awesome option for Infect since the lifegain is irrelevant to your path to victory.
Epic Confrontation is Green's best option for creature kill since in most scenarios it doubles as a small pump spell to help your team crash in.
Serene Heart is an instant speed aura sweeper.
Quiet Disrepair is a neat modal effect, either as slow artifact removal or as Pauper's best recurring lifegain card.
Relic of Progenitus is the default best graveyard hate card in Pauper and it slides into any deck. But Black decks may prefer Nihil Spellbomb to save their own graveyard as a resource. Two other options are the uncounterable duo of Faerie Macabre and Bojuka Bog.The former is a free, instant speed double Cremate and the latter can be re-bought with a Boros Garrison or a Dimir Aqueduct.
Gut Shot is "free" removal, usually for Green decks since they don't have access to Lightning Bolt. It's a great answer on Turn Zero to an opponent's Delver of Secrets, or later on when you're tapped out.
Serrated Arrows is great technology against both fliers and Elves but also Tireless Tribe since a single counter will kill the Tribe at end of turn due to state-based effects. It can also be re-bought with a Kor Skyfisher or reset with a Ghostly Flicker.
Apostle's Blessing is cheap protection for combo decks like Heroic, Izzet Blitz, Tribe or Infect. It can protect against any color removal or from Serrated Arrows, and against many decks, picking one color of protection is the same as granting unblockability.
That's 80 different sideboard cards that you can consider, either as you pick up a popular decks or brew your own. I hope this toolbox was a helpful reference guide to you.
Keep having fun out there,