Hello tentacle worshippers, it has been a long time since I posted here. I've been trying out the new items on Illaoi, and here are my thoughts. Hopefully this helps anyone wondering where to start with the new items, and I would love to hear what you guys think is fun/busted on our favorite priestess.

ELO
For those wondering, I am currently mid-plat, peak rank D3 in season 6 where I was ranked top 10 Illaoi mains on leagueofgraphs. Quit league for two years in college then came back recently. I have been an Illaoi main since she released and I think I have a reasonable understanding of how she interacts with the game :) Anyways, the items:

Tri force
The first mythic I tried, and easily my go-to. While the other mythic items provide good tools for dealing with certain scenarios, Trinity Force is best-in-slot for when you are facing a well-balanced team comp. My simple reasoning: the stacking passive. At max stacks, Tri force increases your base AD by 30%, which is about 50 free AD at level 18. Illaoi has the best base AD at 18 stat in the game, and her scaling on her passive (slam) is stupid high. The downside to this mythic is the obvious lack of durability in both stats and healing that its counterparts provide.

Divine Sunderer
This item is good. It feels really, really good against tanks. Four item Leona? Four W procs and they're gone. The %Armor Pen from the passive synergizes naturally well with Illaoi's AoE slapfest, and the haste and health stats are nothing to shy away from either. However, this mythic loses its luster if fights are over fast, or the enemy team isn't all that tanky. I'm still glad they added this item, it feels like a better version of black cleaver rush in seasons past.

Goredrinker
This is the item for those brink-of-death Tentakill montages moments we play Illaoi for. If you're up against someone with heavy burst or just like playing at the edge of faith, the active item is a saving grace when you wish you had, like, just a little more health to get that second W off in a teamfight. Paired with the Steraks/Gargoyle's combo or a Hydra, its actually a very reliable sustain build.

Frostfire Gauntlet
I really love building this on Cho Gath and being a big boy, but I don't think its for Illaoi. If you're into being a nuisance in teamfights and providing good peel for your carries, this can work, though I prefer to build Serylda's and do the same thing but be the carry. Disclaimer: I haven't tested this item thoroughly, so perhaps I'm missing a good build/playstyle, but Illaoi's winrate is in the hole on op.gg and this is the suggested item, so my point stands.

Ravenous Hydra
Why Riot decided AoE spells should have an AoE disc attached to every hit, I'll never know. It's super funny to me seeing minon waves get slapped with this item, but visuals aside it's actually insane. This item is essentially a higher damage, higher sustain version of last season's deaths dance (more similar to the original 80 AD DD, if any remember). I default to this as my second or third item unless I'm building for more resistances or health stats.

Deaths Dance
Speaking of, we come to this season's deaths dance. This is a far more situational item now as it is only armor, and has no omnivamp. This is good to build into a double AD Top/Jg matchup as a second item, or if a team is full AD, otherwise I would stick with Hydra if you're just looking to elongate your fights.

Sterak's Gauge
Good ol' Steraks. Still a good item on Illaoi for the stall when you're trying to get your ult combo off, though no longer a must-buy, for reasons similar to why I think Triforce is so good. Old Sterak's increased your base AD by 50%(!) which made it turbo-value for both Illaoi and any Triforce user. However, that passive is dead and gone, so I usually only build this item in tandem with Gargoyle Stoneplate for a dumb heavy shield in the middle of a fight. The sustain from the Rage passive is nice when you can spam poke against a Maokai or something, but I think there are still better options for that.

Gargoyle Stoneplate
This iteration of GS is actually viable on Illaoi now, especially with Steraks as mentioned above. The previous version reduced your damage, so most people avoided building it on damage dealers, but there's really no reason not to build this now if you're looking for tank stats.

Muramana
Strange item, I know, but I would like to bring to your attention that all spells proc the shock passive now, including Illaoi's Slam and W. I tested this a lot and really like it, though I wouldn't build it all the time in ranked. If you're playing against a team of all squishies, or think you're gonna run out of mana frequently, buying tear as a starter item and finishing manamune after your mythic honestly isn't a bad idea. This synergizes really well with Trinity and Hydra together, and can be used to make Frostfire a more viable build. At max rank, this adds about 70 extra damage to your hits.

Serylda's Grudge
God I love this item. This solves (not completely, but good enough) Illaoi's biggest weakness: running away. This is a better version of frostfire imho, and I would even go this item third or even second into tank teams or Singed, because of its last whisper component. I am so glad Riot added this item, it actually restored my will to play league and grind Illaoi games back to diamond.

Black Cleaver
Once a core item on Illaoi, I believe this weapon has fallen by the wayside. It's obviously still viable and gives very good stats, but I think the tank shred item slot was stolen away by the Divine Sunderer. Still build this item into a tanky team, especially if your team is AD-heavy, but against normal team comps I've had less success building this item than something more sustain- or durability-focused second and third.

Main Takeaways
There is no single best default item that Illaoi should build. This was one of Riot's goals for all champions with the new item system, and I think it reflects very well here. Each item provides its own wincon and playstyle, much like the Conqueror vs Grasp or E max vs Q max debates.

Out of all the champions I have seen so far, I think Illaoi benefited the most from the new item system. She was a niche character that was the best noobstomper in the game, but got outclassed by anyone who was better than you and read her abilities. Now, she has a range of tools to choose from that I think can allow her to be the machine we all hoped for.

This is by no means a complete or comprehensive list; I have not yet tested all the items that might be fun or viable on Illaoi (Spooky Duskblade, looking at you). However, I have a long history with this champion and I feel like I have a grasp on what the high-elo meta items will be for her, and thought I would share my thoughts here. I would love to know if this helps anyone, or if you have something fun I should try :^ )