To fit her identity, I think she should be an Assassin, and I think she does fit that to some extent in the early game, but she lacks the power to keep that up throughout the game with her kit. She doesn't have the mobility to be an effective assassin nor any inherent durability outside the jungle.
I want her to be an effective AP Bruiser, but the itemization kinda sucks for that. Especially for a jungler. I think she needs to have slightly more durability to make this possible (more HP/lvl/ or armor/lvl?), but it is difficult to put this into her kit without breaking her. In my experience, Elise usually DOES build one or two damage items and then health and resists. Her AP scaling is pretty terrible though right now, and I always feel like I have to sacrifice a lot for items that are not optimum on her just because I wanted some M-pen so I could do meaningful damage.
If the Abyssal rework ever makes it to live, I would like to see Elise pick it up: 
would end up giving her about 1350 bonus health (with HP/level seals), 100 Armor, and 60 MR as far as tankiness, and she would have a decent amount of AOE damage on top of 120 AP and MRshred/M-pen to further increase her base damages. With the build currently, you have 30 more AP at the cost of 300+hp and 10MR. 300hp and 10MR is equivalent to 1000 gold worth of tank stats, and 30 AP is only 652 gold's worth of stats that aren't very useful on Elise while still getting (conditionally) all of the Mr reduction and Magic Pen you used to with some strong passives to boot. She could be a really good Tank buster with this build.
I really like her current kit as far as how it feels to play, though I think her R has a little too high of a CD (lower it to 3 seconds to make her kit feel more fluid). I do think she should maintain a somewhat strong early game, but she should have some more to contribute if she is unsuccessful in the early game. What that entails would be readjusting some of her power from her Q's to her W. I like the idea of giving it lower %HP base damage ~6% (from 8%), but upping the scaling to 4% per 100AP (from 3% per 100AP); this would make it so she would not be as oppressive early but have a little more power if she builds some AP. By reducing the damage on Q, one can add an AOE Slow to Human W and give her some actual in-fight sustain as the recent "buffs" are only effective in the jungle and don't help her fighting capability as much as the old Spider W active. This coupled with some stat boosts will make her tankier builds more worthwhile which I think is the only way she can be super effective with her current kit. If she were to get a complete overhaul to her kit, a more AP stacking assassin build might be possible, but with her current state, I don't think this is possible without overloading her more than she is or her just being completely ineffective as an AP assassin.
That's my opinion. I know it might not be shared by everyone, but I'd appreciate it if you read this and shared your thoughts on this matter. Good job to the OP for opening up a discussion about what people think that Elise is or should be right now. It is an important topic as a lot of people disagree on what Elise is supposed to be. I think it is important that people realize that Riot probably don't have the resources to rework such a new champion (relatively new at least), and that we will mostly have the same kit to work with, so we can only move around numbers and add small effects. I think that with her current kit, she cannot be an effective Assassin after the early parts of the game, but her AP Fighter build is possible to improve upon with some small changes to the champion. Peace.
TL;DR: I want her to be an AP Fighter/AP Tank as I think it's the only thing that can be improved without breaking her/overloading her or her being completely ineffective. New items can help her out here such as the PBE Abyssal that has been "iceboxed", but some changes to the champion herself would still need to happen to redistribute some of her power to compliment this style.
These champions also have much more reliable mobility abilities than Elise's Rappel. Can we find an identity for Elise? What do we want her to be?
or
etc. Elise doesn't really seem to fit this, but lack of durability isn't why.
who has a similar indecisive mix of a little bit of tankiness, single target CC but nothing to write home about, some damage but not really 100-0 burst... but when he lays down that ult people know it.
, 
, 
