What does it take to make a custom dropdown accessible?

Start by trying a native select, since you get keyboard support, focus management and screen reader behavior for free. If the design rules that out, implement the listbox or combobox pattern properly: manage focus yourself, support arrow keys, Home, End, Escape and type ahead, wire up aria-expanded, aria-activedescendant and the correct roles, and return focus to the trigger on close.

My first move is to push back on building one at all, because a native select gives you keyboard handling, mobile pickers and screen reader announcements with zero code, and most designs can be met with styling around it. When we genuinely needed a custom one, for a multi select with avatars, I built it against the ARIA authoring practices listbox pattern rather than improvising. That meant the trigger is a real button with aria-expanded and aria-controls, the popup has role listbox with options that have role option and aria-selected, and I track the active option with aria-activedescendant so focus stays on the input. Keyboard wise you need up and down, Home and End, Escape to close and restore focus to the trigger, Enter or Space to select, and type ahead to jump. The important thing I tell people is that ARIA only announces state; it adds no behavior at all, so a div with role button and no key handler is worse than useless. I test with the keyboard only, then with a screen reader, before it goes near a review.