(not a native speaker)
I have to agree that her W feels rather weak overall.
While I think it's good to give her at least some base damage there (for finishing off enemies, for instance), in my opinion one of her main weaknesses doesn't lie in not having high enough damage numbers. Her weakness lies in applying these numbers to someones face, which can be alternately extremely frustrating or extremely overpowered.
One of the main problems: It's really hard to make a spawned tentacle hit a perceptive player (especially problematic in higher elos), which is absurd in that sense that the tentacles are meant to create a zone that is supposed to be a damage threat. Since her chunky W and the slow tentacle animation telegraph the damaged area like 2 seconds in advance, it's nearly impossible to apply damage in some scenarios.
I'm fine that champions can dash out of danger, but it's really frustrating that it is possible to simply walk around all of her damage.
The solution in my eyes is to tie the increased slap speed to her W instead of her R. The speed of Q and E seems balanced, but on the W it's just really underwhelming.
This would make it easier for Illaoi to play around her tentacles more and it would be pretty fair as well, since walking up to a tentacle when illaoi is around should hold some reliable damage threat.
It think that change would especially have some positive effect in higher elos, where she often can't hit stuff at all.
As a side note, I also think the slow on her E should be applied upon hitting the tentacle instead of when the tether breaks. This would make her by no means a cc machine or particularly oppressive, but give her at least some catch potential and team utility and add a little more depth to her possible plays. As it stands now, the slow is completely useless in most of the cases (except when multiple opponents chase the enemy across the entire map).
From my perspective, these changes would address her weaknesses without simply increasing numbers or reducing the tentacle theme.