For a start, Allium is a genus and not a family. Wikipedia describes it as "the onion genus", so you could argue that "onion" was a perfectly acceptable answer to this question.
The family that the Allium genus belongs to is the Amaryllidaceae, or (in English) the amaryllis family, so either of these is actually the correct answer.
When setting questions about scientific classification of species, we should bear in mind that the hierarchy is basically Kingdom, Phylum, Order, Family, Genus, Species. I have a mnemonic for this, which I have found indispensable: King Philip Caught Our Father Going Swimming.
In 1990, the American microbiologist Carl Woese introduced a new level at the top of the hierarchy: the Domain (a.k.a. Superkingdom or Empire). There are three domains: Archaea, Bacteria, and Eukarya. Anything you're ever likely to get asked about in a quiz, in any detail, is included in the Eukarya. (For more information on this, check here.)
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