Lissandra fills the role of the ice mage, brimming with movement reducing crowd control (slows and snares) and bringing plenty of burst to the table. Her cool downs are generally fairly low (particularly on her Frost Shard). Her biggest reasons for bans are due to her fairly good laning prowess (especially in top lane), her ability to completely lock down highly mobile champions commonly prevalent in high level play, and her invulnerability (5 seconds of invulnerability with a Zhonya’s).
Lissandra’s primary identity is her crowd control utility, with a secondary identity of burst mage (the “ice mage” theme between the two). Her tertiary identity is her invulnerability. Changes to her should hopefully avoid cutting her cc utility and damage. Her ultimate might get hit, but it has a lot of innate skillful play to it. If she wants to make herself invulnerable, she loses out on her only hard cc. If she uses it offensively, she no longer has 2.5 seconds of invulnerability. The right decision can make or break her in a fight.
Her strengths lie in her ability to lock down an opponent with a lot of crowd control abilities and her ability to kite. Her weaknesses are her shorter range on her abilities (her Q is around average for mages, her W and R are lower), the ability to stop her escape through snares/stuns and her low survivability early (one of the lowest base HP and Armour, in addition to lowest base move speed).
A few proposals for alterations to her kit, with all this in mind:
1) Reduce the range on her Q. It makes her slightly less of a lane bully, especially in top lane, solving one of her issues. It pushes her to get in closer during team fights, forcing her to be a little more cautious. It also amplifies her need to kite well, to keep people from standing on top of her and killing her. Her identity isn’t changed, her strengths are amplified (a little) and her weaknesses become a little more prevalent. Issues would be that it would make her more difficult to lane with in mid lane. She might also have to place herself into harm’s way and would be too easy to catch and blow up. The reduced range might hurt her ability to kite successfully, completely removing one of her strengths.
2) Reduce the speed of her E. Reducing it gives opponents more time to react to her dash, increasing the window of opportunity. It enhances her weaknesses and, again, increases her need to kite successfully. One issue would be that the dash wouldn’t increase total distance that much – she could walk nearly as far as the dash would move her if the speed is too slow. Another issue might be that it wouldn’t really alter any of her issues, as she still has high laning prowess and lock down in a fight.