Probably the best explanation comes from the Hufflepuff welcome letter on Pottermore, which makes several comparisons between the house traits and badgers:
Hufflepuffs are trustworthy and loyal. We don’t shoot our mouths off, but cross us at your peril; like our emblem, the badger, we will protect ourselves, our friends and our families against all-comers. Nobody intimidates us.
and a bit later:
You will hear other houses boast of their security arrangements, but it so happens that in more than a thousand years, the Hufflepuff common room and dormitories have never been seen by outsiders. Like badgers, we know exactly how to lie low – and how to defend ourselves.
When Voldemort encounters Hufflepuff’s cup in Half-Blood Prince (while Harry and Dumbledore are using the Pensieve), he finds it has an engraving of a badger.
“A badger,” murmured Voldemort, examining the engraving upon the cup. “Then this was…?”
“Helga Hufflepuff’s, as you very well know, you clever boy!” said Hepzibah, leaning forward with a loud creaking of corsets and actually pinching his hollow cheek.
As far as I know, canon doesn’t tell us when this cup was made, so it’s possible Helga Hufflepuff had a familial connection to badgers. But that’s speculative at best.