I've read on Bulbapedia that the logo is a "stylised 'L'", however I feel like it may have a deeper meaning.
What does the icon on Ash Ketchum's (first) hat mean?
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Per the Pokémon wiki
Ash's first hat, worn during the original series, is easily the most well-known: an official Pokémon League Expo hat that was red and white with a green, stylized "L" on the front, which Ash claimed to have "sent in about a million postcards" in order to obtain. Misty seems to have tried to win it as well, as she stated she sent in one postcard. Ash wore this hat during his journey through Kanto, the Orange Archipelago, and Johto, and was very attached to it, becoming enraged when a Mankey stole it and played keep-away with it in Primeape Goes Bananas. The design may be loosely based on Red's Generation I hat, and in early concept sketches for the anime, Ash's hat matched Red's exactly.
From Primeape Goes Bananas (25th episode of the first season)
Misty: I didn't know that was an official expo hat. I remember entering the same contest myself.
Ash: I had to send in a million postcards to win that hat!
Misty: He he. No wonder I didn't win, I only sent in one.
Interestingly, the hat went through several iterations before the concept artists finally settled on the "L" symbol. Unfortunately, my Japanese is insufficient to decipher the text on the right hand side at the bottom.