Instead of re-creating the store every time name changes, only create it once and set $player.name when name changes.

<script>
    import {editableStore} from "./store";
    
    let name = "John";

    let player = editableStore(name);
    $: $player.name = name;
</script>

This will require you to update your store method to return the set function.

export const editableStore = (name) => {
    console.log("Recreated with name:", name);

    // also destructure set here
    const {subscribe, update, set} = writable({...defaultStore}, () => () => clearInterval);

    if (name) {
        update(s => ({...s, name}));
    }

    const clearInterval = setInterval(() => {
        update(s => ({...s, log: [...s.log, new Date()]}))
    }, 1000)

    // also return set here
    return { subscribe, set };
};

Try to instanciate your store as son as possible like in the ./store.js file and then use the set or update method instead of instanciate it in the component directly:

// store.js

import {writable} from "svelte/store";

const defaultStore = {
    name: "Bob",
    age: 18,
    log: []
};

export const createEditableStore = () => {
    const {subscribe, update, set} = writable({...defaultStore}, () => () => clearInterval);

    const clearInterval = setInterval(() => {
        update(s => ({...s, log: [...s.log, new Date()]}))
    }, 1000)

    return { subscribe, set, update };
};

export const player = createEditableStore()

<!-- App.svelte -->

<script>
    import { player } from "./store";
    
    let name = "John"

    $: player.update(p => ({ ...p, name }))
</script>

<h1>Hello {$player.name}!</h1>

<button on:click={() => name = (name === "Bob" ? "Jerry" : "Bob")}>
    Change name
</button>

<h2>Log:</h2>

{#each $player.log as log}
    <li>{log}</li>
{/each}

Have a look at the REPL.