As always, I will contribute my own thoughts as well.
Character Design
There are 2 major groups when it comes to his visuals: some people think he is a young mummy boy (so, he was a human once) and the other group says he is a Yordle.
I belong in the second group, I always imagined Amumu to be a yordle and in fact, I want his design to clarify it much more.
There is a well known concept for his VGU, made by Andres Castaneda for Riots art contest.
https://us.v-cdn.net/5021068/uploads/editor/7e/cl1p3ndi6yhv.png
In my opinion, it is almost perfect to go. It shows that he is a Yordle, even if it can be a bit more mysterious in lore. As a Yordle, he has glamour, meaning that the people of Shurima might still see him as a young boy.
It has great details, something that current Amumu is absolutely lacking. He is just bandages and eyes, one of the most boring champion designs in the game right now.
Giving him some golden jewelry (for which Shurima is known for) and these big Yordle ears really made the concept for me. And I think that he could wear his bandages more lose, just like here. There is no need to hide his whole body behind bandages.
His depression theme is iconic, but I think it is overdone for him. He doesn't have to cry all the time, they can make him desperate and depressive in his VO. Just make it more serious. I don't think that an always crying little boy is really something alot of people enjoy when it comes to thematics.
Splashart
Something that I really want to happen for his splashart is showing him with an actual sarcophagus.
I always wondered why this isn't a thing from the start, since this is something highly associated with mummies in media.
Please show Amumu resting lonely in his Sarcophagus, reaching out a hand.
Maybe with some green spooky mist. ;)
Gameplay
It's hard to imagine something for Amumu when it comes to gameplay.
I think the most iconic parts of his kit are his Q (Bandage Toss) and his R (Curse of the Sad Mummy). These 2 skills would need some work, of course. His ult could be bandages slowly pulling enemies towards Amumu, instead of an instant root and nuke.
Also I would like to keep his magic damage enhancing theme.
The crying and tantrum never really convinced me as parts of his kit.....
He might be sad, but he doesn't really have to kill his enemies by crying them to death. It is too cliche playing into his depression theme in my opinion.
Something that current Amumu is clearly lacking is a tank steroid. He just has flat damage reduction. Compared to the tank steroids other champions have, it is pretty bad. Maybe they could even build in the sarcophagus from his splashart in some way? For example, if he could have a mechanic, that let's him enter a sarcophagus and rapidly regenerate health in it.
Lore
I would like to investigate the cause of his curse in the lore a bit more. Was there a tragic event in his past that cursed him?
In the video The Curse of the Sad Mummy, there are these shadows haunting him and telling him that he will be alone forever. Maybe they could build these in in some way?
Could the people who he killed with his curse haunt him as ghosts, permanently reminding him that he is a monster?
Kinda reversing the loneliness, he is never alone, but these ghosts always just tell him that he shouldn't exist in the first place and always just brings harm.