Eating in Hanoi is mostly about the street. The city's signature dishes — pho bo (beef noodle soup), bun cha (grilled pork with rice noodles), banh cuon (steamed rice rolls), cha ca (turmeric-marinated fish with dill) — are almost always better from a specialist street stall than from a sit-down restaurant. Most places open for one meal a day and close when they run out.
A few streets and institutions to know. Cha Ca La Vong on Cha Ca Street (Old Quarter) has been serving its one dish, turmeric fish, since the 19th century. Bun Cha Huong Lien is where Obama and Bourdain ate; the resulting "combo Obama" is a reliable tourist magnet but the place is legitimately good. Pho Thin on Lo Duc is the classic old-school pho counter. Banh Mi 25 is the go-to for the Hanoi take on the French-Vietnamese sandwich.
For a proper sit-down dinner, La Badiane, Gia Nguyen, and Home Hanoi (all in or near the French Quarter) do elevated Vietnamese.