Does Halo 3: ODST Have Custom Games?

No, Halo 3: ODST does not include a Custom Games lobby. Unlike Halo 3, you cannot create private matches with custom rules, maps, or settings within ODST itself.

What Multiplayer ODST Does Offer

• Firefight (Co-op Survival): You can select difficulty, map, and a few skull modifiers, but you cannot invent new game types or alter core rules (player count, weapon sets, win conditions).
• Matchmaking: Standard matchmaking for Firefight and the included Halo 3 multiplayer maps (via the second disc / "Mythic" map pack).

The Workaround: Halo 3's Forge & Custom Games

The Halo 3 multiplayer disc included with ODST (or the standalone Halo 3 game) does have Forge and a full Custom Games suite. You can build maps/modes there, but they are only playable in Halo 3, not the ODST campaign or Firefight.

If you want a deep custom-game experience with the ODST aesthetic, you'll need to play Halo 3 proper or look to Halo: Reach / Halo: The Master Chief Collection on PC/Xbox, which include ODST Firefight with full custom-game support.