Preface: At one point in time, all of this was neatly arranged but changes to the editor and code and whatnot have been made since then so it's all a bit off now. I can't wrap my head around it. You fix one spacing issue and another appears; it is what it is.
I can't believe I've legitimately written this, but here we go.
Boba first appears in The Empire Strikes Back (ignoring the prequels, brief added scene in A New Hope in which he looks directly into the camera - for the time being - and the Star Wars Holiday Special).
His first scene sees him amid a plethora of other bounty hunters wherein Vader singles him out and says
"I want them alive. No disintegrations." This hints at a history between him and Vader (not to mention the history implied by his banged-up armour) and that he's more dangerous and reckless than the others in attendance.
There's also, of course, the implication that he's the one who dealt with Owen and Beru - intimately tying him to Luke. He replies with a cool, collected "As you wish." If Vader hadn't specified that rule, you just know that Boba would've literally obliterated Han and the gang.
The next we see him is during the scene in which the Falcon attaches itself to the Star Destroyer in an almost successful attempt to lose the Empire - by confusing their tracking. It's only "almost successful" as Boba realises what they've done, his cunning tops Han's. When the Falcon detaches to float away amongst the garbage, Boba's there and stealthily follows them. This moment demonstrates that he's experienced and clever - moreso than Han and the entirety of the Empire.
Boba figures out that the Falcon is heading to Bespin, informs the Empire and they both arrive there before Han and the gang. This leads to the Empire's deal with Lando. Han is tortured to beckon Luke. Lando is visibly distraught at the sound of Han's horrifying screams, but Boba remains unmoved and simply appears to be patiently waiting.
Vader then comes up with the idea to carbon-freeze Han. Boba is at first opposed to this. "He's no good to me dead." He's aggressively talking back at Vader, yet Vader doesn't immediately choke him out as he has done to anybody else who's questioned him. Vader respects him. Vader is only ever shown to have any respect for three people: Grand Moff Tarkin, the Emperor, and Boba.
Following Han being frozen in carbonite, Luke arrives. As he's skulking about Cloud City he comes across Boba and some workers wheeling Han to Slave I. Luke wants to remain stealthy so hides, but then pulls out his pistol. As this happens, Boba whips his head to the right. He heard it, but continues on as if he hadn't. Soon after, Boba unexpectedly pops out and takes shots at Luke, corralling him into the carbon-freezing chamber where Vader awaits.
Despite Leia, Chewie, and Lando's attempts to thwart Boba; the bounty hunter manages to get away with Han. Successfully completing his mission, he heads to Jabba's Palace.
The result of Boba's actions in Empire? Luke's hand is cut off and Han's taken. Essentially the reasons for Empire's ending being so dour. Thirty years before Thanos, here's an antagonist ending a movie having successfully bested its heroes.
Return of the Jedi. Boba's at the Palace, where he's shown to be Jabba's right-hand-man. This is further reinforced by the aforementioned, additional scene added to A New Hope. So that's the respect of both The Dark Lord of the Sith as well as the galaxy's most powerful crimelord. Speaking of additional scenes; he's a lady's man, flirting with the dancers.
Leia shows up, disguised as Boussh, and whips out a thermal detonator. Every single other attendee screams and cowers in fear, but Boba just whips out his rifle. Following the aggressive negotiations, Boba gives Boussh a nod of respect. This is likely on account of her managing to bring in Chewie and/or that they're both badass-armour-wearing bounty hunters. He sees himself in her.
The next we see Boba is immediately after Luke begins decimating Jabba's desert skiffs with his lightsaber. Boba's uses his jetpack to launch straight into the thick of the action. He ties up Luke and attempts to shoot him, but that's thwarted as Luke deflect's the blaster bolt with his sabre and then slices the blaster in half, knocking Boba down. Could you imagine any other non-force-user taking on a Jedi or Sith (besides Finn)?
EDIT: Oh wait, turns out Finn is force-sensitive after all. Only supports my case.
This is probably the first Jedi Boba's seen since the Clone Wars wherein one beheaded his father before his very eyes. Standing back up, Boba writes off the fumbling, blind man (what could he possibly do?) and instead aims at Luke. The Jedi's in the sights of his flamethrower and would've been dead to rights if not for Han luckily managing to swat Boba's jetpack.
This admittedly leads to Boba's unfortunate demise. He uncontrollably spasms his way into the Sarlaac Pit.
In conclusion, Boba is a very active antagonist and certainly more than just a cool suit of armour. He's mysterious, cool, cunning, flies one of the most interestingly designed ships we've seen, and deals in the shady underbelly of the galaxy. Sound familiar? He basically a dark version of Han. The yin to his yang. What Vader is to Luke.
This is ignoring the Expanded Universe (Legends), the animated series (Star Wars: The Clone Wars), and the prequels (for the most part). Not to mention his design, the fact that Jeremy Bulloch based his movements and mannerisms on Clint Eastwood, his gravely and badass voice (not to mention one that's just as cool, that of Temuera Morrison, in the updated versions), as well as the hype that preceded his arrival in The Empire Strikes Back.
He isn't overrated. He's actually cool. And he's certainly more deserving of the popularity than the likes of IG-88, Bossk, and (god forbid) Darth Maul or Captain Phasma. If you believe and tout that he "Does nothing but stand around and look cool before embarrassingly dying," you're flat-out ignoring the subtleties and what actually happened. Case closed. End of story.