Good day Sir @ornicar , I am a graphic designer and a chess player also :). I want to contribute to your project by proposing a new design for Icon/Logo as i have observed and notice the existing icon needs an upgrade in my opinion.
With this issue I have come up with these designs which I hope you will like and if you do, I will give and send you the files as a gift for free.
Here it is Sir:


If you have suggestions or any idea for changes just let me know and i'll edit it.
Thanks and best regards!
-Tobaloidee
Wow, that's very impressive work @Tobaloidee! Thank you!
The team is discussing it extensively and I'll do my best to sum it up here. It's my first time reviewing and commenting a logo, sorry in advance if I sound unappreciative or make the same mistakes I'm sure most clients do, when I'm not even a client.
Now for the discussion and ideas:
Li is a good idea that may be refinable. The L is not obvious at the moment, leaving a very visible i that reminds us of apple products or iChess. Maybe the i could be more subtle? Or the L more obvious? Here's what a team member (@michael1241) quickly drafted: 
L? These are just half-baked ideas for the sake of discussion of course.
I hope this inspires you and makes you want to try some alternatives.
Quality design contributions are the rarest and most valuable thing in Open Source software. We're very excited to land this logo! Thank you!
Hi Sir @ornicar , thanks a lot for the very good feedback..I will take note of those and try to make some alternatives. I will post it again here. :) I will do my best
Hi Sir @ornicar I fixed the shape and I came up with 4 variations. let me now Sir whatever suits the app I will be editing :)

Great ones, wow. What do you think about these alterations of the top-right design? We removed the dot from the i.
Top left is (almost) your original design.

It's great Sir! cleaner and more balanced..that four alteration all looks awesome.
Should I finalize it and send here the needed files Sir? I will make it an adaptive icon also..
Let's not rush it. We'll give it more thoughts for at least a couple days.
Changing logo is a big commitment. It's not only the mobile app we're talking about, it's also the website, the streams, the API signing buttons, the press kit, the merch store... the entire branding. It's going to take a lot of work and time, so we better be sure to get it right.
I fully understand Sir! All for the best of Lichess..
We kept on simplifying the design, eventually dropping the Li because it doesn't scale to small icons. The result works in 14x14px:


Still working on it.
Nice Sir..orange color is also the brand color Sir? Im working also here..:)
Actually Sir I also thought of that hidden big letter "L" which is on the neck part is on my first layout.. its just hidden :)
Lichess doesn't have a brand colour yet, but I'm thinking of using this orange #d85000. We've been using it for years as a highlight colour on the website.

That's a nice color to combine with white or black. here Sir i came up with these
I put a shadow for the adaptive icon visibility

sample adaptive icon (mock-up)

I tried to play with the horse shape and i came up with this one Sir. more horselike specially the negative space inside.

Interesting, but I like the former one better. Did I mention I'm a huge fan of simplicity? :D

^ This one is really awesome.
I just want to show you Sir all that I can give to the app..馃榾 I am also a believer that less is more..
that former style rocks! it wil gain more interest for the users to play Lichess
I think we're almost there. There's one last thing that bugs me: the inside shape doesnt look like a horse head.
I don't know if it's possible to improve that without ruining the overall design. What do you think?
I will fix it without changing the outer shape sir dont worry.. :) will show you asap
here Sir i came up with 2 results
-more closer to the previous one (minimal adjustment)

-i made a new shape, we cant arrange the inside shape of the previous one without adjusting the outside shape thats why i created a new shape

This looks really good (the first one is the better one in my opinion)! As a hobbyist designer myself, this rebranding excites me! I'm just commenting to make sure that this thread isn't forgotten. Also, if there's anything I can do to contribute, please to let me know! I've been looking for ways to help lichess for a little while actually.
One thing that has come to mind is the chiseled nature of the nose in the first logo. To me, it kind of clashes with the nature of the rest of the horse a bit. Can you see what I'm saying here, or is it just me? Most of the horse is made with elegant curves and long sharp lines, and the nose breaks this structure.
I quite like the below rendition.

If I had any commentary, it would be to try tapering the back spine similarly to the bottom line. i.e. gradually lower the weight of the line from the top of the ear down to the bottom of the back.
Edit:
Actually, this one is probably better. But same comment. I'd like to see it with a thinner spine.



How is it going?
Hello, sorry for the long delay, we were busy and fell behind but we haven't forgotten you and are still discussing some details. We should get back to you soon!
Goodmorning, still no news?

As mentioned by @ornicar it is fitting to the Lichess GUI and the minimalistic style.
The choice of color:
Orange: Familiar because of lichess
White: Color of Freedom --> See Lichess Philosophie
The other ones look a little chunky (Personal preference?)
The knight looking to the right is better, because in Europe reading direction is from left to right. There is a lot of art theorie behind this.
It looks more similar to the old logo, therefore it would meet expectations of the community.
Warning: I am not a graphics designer or even close to it. This is just so that you can get an idea.
_L_ and _i_ would be readable and the logo will remain its full power.

@Tobaloidee thank you so much for your effort, speding your free time on a non-profit project should be honored. Appreciate it!
@ornicar Is a potential new logo still relevant? Not much going on here ;(
@ornicar Just bumping this thread.
Thanks @Stapelueberflieger 馃憤
Why was the thread closed? Due to low participation or has it become irrelevant?
It has been a year.will just reopen it when they become available Sir.
Most helpful comment
Greetings, Lichess Community!
This one is neat!
As mentioned by @ornicar it is fitting to the Lichess GUI and the minimalistic style.
The choice of color:
Orange: Familiar because of lichess
White: Color of Freedom --> See Lichess Philosophie
The other ones look a little chunky (Personal preference?)
The knight looking to the right is better, because in Europe reading direction is from left to right. There is a lot of art theorie behind this.
It looks more similar to the old logo, therefore it would meet expectations of the community.
Just another Idea
Warning: I am not a graphics designer or even close to it. This is just so that you can get an idea.
_L_ and _i_ would be readable and the logo will remain its full power.
@Tobaloidee thank you so much for your effort, speding your free time on a non-profit project should be honored. Appreciate it!
@ornicar Is a potential new logo still relevant? Not much going on here ;(