What is the meaning behind the Joker gaia memory logo ?
 

Like, why it's flames and lightning ?
Is it supposed to mean something ?

I think the vagueness was an attempt to reference the wild card, the Joker.

I don't believe it's flames or lightning specifically, just energy.

Joker's ability is enhanced physical prowess. Whatever memory it combines with enhances W's body*, such as giving Luna stretchy limbs, or Heat fire punches. Kamen Rider Joker likewise gets enhanced attacks with energy culminating on his fists/feet, and speaking esoterically, the Joker memory could embody the concept of ki/chi, which is energy within the body focused and manipulated. The Joker symbol could therefor be a J made of ki energy.


*Side note: as much as I like the W manga, I hate how it fundamentally misunderstands how W's form combos work. FangJoker generates blades from the body because it's combining Fang (blades) and Joker (body) together. FangMetal and FangTrigger ought to augment their respective weapons, since that's what Shotaro's other memories bring to the combos, but instead it just generates different body spikes/blades. Following W's combo mechanics I'd have expected the Trigger Magnum t have a bayonet, or the Metal Shaft to be a Naginata/halberd of some kind.

Myself personally, I was thinking a court jester's costume. It's not obvious, but I always felt whoever worked on the graphics tried to express a court jester costume in an abstract way in relations to the whole "joker" theme.

It's possible that the Joker's recollections are a manifestation of ki/chi, the channeling and control of one's own internal energy.

To expound on this, it's the shoe/boot, specifically.

Then it's a funny kick. Explanation of the theme of the show.