I'm a balanced blend of #3 (focus on the role-playing part to boost their impact) and #6 (increase everything: stats, items, time planning, etc).
Generally, it depends on the source of the monster. Monster Manual? I lean toward #6.
Volo's Guide or Tome of Beasts (Kobold Press)? I lean toward #3, as those monsters feel a bit more robust.
Often, it depends on the context in which I'm using the monsters, the size/power of the PC party, and the particular monster. For instance...
Recently ran a fight with a seven-headed pyrohydra (CR 13) vs. 6 11th-level PCs and supporting NPCs. Mechanically it was a souped-up hydra (CR 8) with fire-breathing and legendary actions (a reactive tail slap & using its breath to cause pool to boil), but I also hid it in an oasis pool, allowing it to surprise some PCs / all NPCs and drag grabbed PCs underwater in its jaws/coils.
Lore-wise, I made it an ancient monster summoned by evil fire mages to despoil the oasis & discourage anyone from rebuilding after their raid. The PCs found a cracked brass puzzle-sphere at the bottom of pool that they rightly suspect was expended (like a scroll) to summon the pyrohydra. They may later learn it was summoned from the Straits of Varrigon between Planes of Fire and Water (Plane of Steam), which may be relevant as their adventure will take them to the Inner Planes at some point, and they could visit the Straits of Varrigon, e.g. as a sneaky route to sail into the City of Brass undetected.
There's another layer of lore, with similar engraved metal puzzle-scrolls being used by elven desert tribes in the setting, hinting that the elves may have been the ones to first bind pyrohydras...but somehow the evil fire mages got that knowledge.
While it very much felt like a "throw away" encounter initially – a fight to blow off steam & break up lots of interaction/exploration – the more the players investigated the more there was to discover about the creature, adding needed depth to an otherwise shallow experience. The mechanical changes helped it survive long enough against such a potent party in order to use its powers and provide a semblance of threat.