Heliopolis is part of Cairo, and Cairo is full of things worth spending a day on.
Egyptian Museum and Tahrir Square — About 30 minutes from Heliopolis by metro (change at Attaba or Sadat). The Egyptian Museum on Tahrir Square holds the world's greatest collection of ancient Egyptian artefacts, including the Tutankhamun treasures. Downtown Cairo around the museum has its own character: crumbling belle-époque buildings, bookshops, and coffee houses.
Coptic Cairo — About 45 minutes south by metro or taxi. The oldest part of Christian Cairo, with a cluster of ancient churches including the Hanging Church (built over a Roman gate), the Church of St Sergius (reputedly built over the site where the Holy Family sheltered), and the Coptic Museum. One of the more quietly remarkable places in the city.
Giza Pyramids — About 45 minutes by car or taxi (the metro does not go to Giza directly). The pyramids of Khufu, Khafre, and Menkaure, the Sphinx, and the vast plateau they occupy are the obvious reason many people come to Egypt at all. Go early in the morning if possible, before the heat and the crowds.
Khan el-Khalili — About 30 minutes by taxi from Heliopolis. The great medieval bazaar of Cairo, centred on the al-Husayn mosque and spreading through a labyrinth of covered alleys. Touristy in parts but substantial and genuine underneath: the spice market, the goldsmiths' quarter, and the coffee houses have been here for centuries.