{
            name,
            ...
          }:
let
  ihnerit (config.flake.modules.x10.device-config.${name}) username;
in
          {
            options.homeManager = {

          {
            name,
            ...
          }:
let
  inherit (config.flake.modules.x10.device-config.${name}) username;
in
          {
            options.homeManager = {

Something I'd already tried earlier but discarded due to an unrelated error (that I thought told me I could not do things this way):

{
    type =
      with lib.types;
      attrsOf (
        submodule (
          nixosConfig:
          let
            cfg = nixosConfig.config.nixos;
            inherit (nixosConfig.config) hostname username system;
          in
          {
            options.nixos = builtins.break {
              entryPoint = lib.mkOption {
                type = lib.types.path;
                default = "${repoRoot}/nixos/${username}_at_${hostname}";
              };
              toplevelBuild = lib.mkOption {
                type = lib.types.raw;
                readOnly = true;
              };
            };
            config.nixos.toplevelBuild = lib.mkIf (builtins.pathExists cfg.entryPoint) (
              inputs.nixpkgs.lib.nixosSystem {
                modules = [
                  {
                    nixpkgs.hostPlatform = { inherit system; };
                  }
                  (import cfg.entryPoint (inputs // { inherit repoRoot; }))
                  "${repoRoot}/x10/options/unfree-packages.nix"
                  inputs.nur.modules.nixos.default
                ];
                specialArgs = {
                  inherit inputs hostname repoRoot;
                };
              }
            );
          }
        )
      );
  }

So yeah, the inner config variable of the extending submodule contains the submodule's "sibling" option values.

x10an14: taking the discussion from the beginning …

If you want to declare options specific to one instance, you can do it as follow:

{ lib,  ... }@toplevel:
{
  config.flake.modules.foo.<fooInstance> = {config, ...}@submoduleInstance: {
    options.home-manager = lib.mkOption { ... };
    config.home-manager = ...;
  }
}

If you want to make the home-manager option available to all instances, then your initial example is almost there, you have to redefine the foo option with the submodule that you want to add. Submodule option declarations are merged, see fileSystems option as an example, which is declared in multiple modules.