You've got me thinking of all my past projects and what the first bosses were like. I'd have to say just a part of the story, better stats. I've since looked into a little more and I think that I still have them remain a big part of the story, and those higher stats, but challenging the player or introducing something else that may be coming up. A way to foreshadow things. Whether mechanics, a new set of items, maybe a level with puzzles and just little nods to it.
Since I haven't made any bosses in my current game yet (it's all world construction and quest flow right now) I'll mention one I made in a shelved game--one I intend to put a ton of effort into at some point since the characters were based on an old Flash game of mine I wanted to revisit. Instead of the Rat Lord, Giant Bee, or Slime King, you take on the Ultimate Big Bad. Or rather, the main character is daydreaming that she's some sort of chosen-one badass, and that another character she has some rivalry with is the Ultimate Big Bad. Since it's a daydream, all the attacks were intended to be flashy and over the top instead of "Rat Lord uses Poisonous Bite." It was fun to make, but looking back at it now, I've learned a lot about manipulating action sequences and abusing forced actions to the point where it wouldn't measure up to my current standards. Still, when the time comes to revisit it, I'm sure it'll be loads of fun to work with.
Well, the first boss fight in my latest game is rather a "fake boss fight" for tutorial reasons. Let's say the first real boss fight.. In order to make the boss visually stand out, it is much bigger than normal monsters + uses dragonbones. Auditive it has its own battle bgm with variations on certain stages (when HP reaches a certain point). I like bosses appearing epic. Yet the first boss is quite easy still. Def + Atk not too high, rather make it a HP sponge. No new skills, but rather a combination of skills, normal monsters have used before.
First boss of first actual project was a demonstration of the game's Taunt mechanic, mainly that you need to have one person defend against a powerful attack. Current project still in the alphas, but thinking about it, it should be a showcase of that game's main mechanics: The first boss is a Door, but two untargetable Walls can deal left/right AoE. You would need to have your leftmost and rightmost members Guard to prevent those AoEs from hitting your inner members, plus you should try to AoE so that you can damage the walls as well. You may also want to swap members to make healing less frequently.
My first boss is an orc who has a large retinue with him. He has lots of stun attacks and summons reinforcements as he becomes damaged, but he's susceptible to confusion and attraction, so can be used to kill his allies.
I have more.... minibosses.... than 'bosses,' persay, because the ones I implemented so far are exploration rewards. The first one is in an area where the standard enemy mostly does nothing but apply annoying status effects and waste turns. The encounter starts with the basic enemy visible, then reveals another... and then... a big, bad, NEW enemy rumbles in. Gameplaywise, it inflicts debuffs and MP-drain attacks, which is dangerous for the player because their main damage dealer up to this point has been a mage who regens MP on the regular. It's meant more as a scare and in-universe flavor text than a grand battle; while the player might have to run away the first time they meet it (gaining a warning that it'll aggro as soon as they return to the map) when they return a little bit afterwards with a new skill it's easy to KO.
My first boss is pretty simple. The party visits an old temple seeking advice from a deity said to reside there. Unfortunately, she's asleep and the golem guarding her mistakes the group for intruders, thus a battle ensues. The battle itself is pretty easy, especially since the player has a safety net in a high leveled character who, for plot reasons, cannot attack, but can use healing items. As for what the Golem can actually do: he uses a mix of physical attacks hitting the entire party and strong magic attacks hitting a single party member; At low HP, he'll summon some Wisps that can also use magic and inflict Sleep.
The first boss in my game is a were-cat named Fiera. A half-human that learned fire magic. You exit a dark forest at night, trying to find shelter from a fierce thunderstorm. She attacks you because she's hungry, blocking your path just as you were about to escape into a cave. I build up the suspense with slow panning camera movement, zooms, and fitting music. There's also some sprite animation of her ears and tail. Here I introduce the charge mechanic where it's a good idea to block.