The House of Sampoerna is the best museum in Surabaya and worth a morning of anyone's time. It occupies a beautifully restored Dutch colonial compound in the old city, and traces the rise of one of Indonesia's biggest tobacco companies from a small hand-rolled cigarette operation to a national brand.
The museum is genuinely interesting rather than simply promotional, with good exhibits on the history of kretek (clove cigarettes), the colonial spice trade, and the role of Chinese-Indonesian entrepreneurs in the city's commercial life. The factory floor where workers hand-roll cigarettes can be viewed from a gallery above. The café in the courtyard is a good place to recover from the heat. Entry is free.