Beifong - Northern Square, 北方 (Mandarin). The name sort of implies that Toph’s family is originally from the north of the Earth Kingdom and maybe got pushed down southward because of the Fire Nation occupation. This would explain why Toph’s family name seems to carry so much weight in the Ba Sing Se area. The “square” part of her name might be a reference to her family’s estate or possibly Chinese/Earth Kingdom coins, which have square holes in the middle. The square is also the shape that tends to represent the Earth Kingdom, as well.
Also, this is the only surname mentioned in the entire series! Unless you count the Fire in “Wang Fire”. Multi-syllable surnames (复姓) are somewhat rare in Han Chinese culture and tend to carry a carry a more elegant or “historical/fantastical" feeling. So the name Beifong is pretty fitting for both Toph’s family and the Avatar setting.
Toph - Tough and/or toff (English). So, as explained in the Ember Island Players, Toph’s name is a corruption of the English word tough. Cause she’s tough. But it could also be a reference to the British slang toff, a derogatory term for someone with an aristocratic background, particularly someone who exudes an air of superiority. Both meanings are equally relevant to her character.
Lao (Toph’s Dad) - Long-standing or outdated, 老 (Mandarin). A pretty fitting name for an old money aristocrat. His views on disability are certainly very outdated.
Poppy (Toph’s Mom)- A type of flower (English). From what I’ve gathered from some questionable floriography (flower language) websites, the poppy represents sloth and death in Chinese culture. Poppy seeds are used to make opium, so that’s pretty fitting symbolism. It’s also an appropriate symbol for a complacent aristocrat who ignores an ongoing war in favor of personal comfort.
Okay, to clarify, the characters for Beifong, 北方, mean something like “northward” or “north direction” when combined, but the individual characters mean “northern” and “square” respectively. Credit to @pauliestorylover for first pointing this out.
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