So, your question is what makes Hufflepuff stand as a house. I would say that a Hufflepuff can be as brave as a Gryffindor, as ambitious as a Slytherin and as smart as a Ravenclaw, the difference between a puff and any other student from another house is that, even if the puff may have any other qualities, the one which he or she values the most is friendship, loyalty,and so on.
Remember those descriptions of the four houses? "Gryffindor/Slytherin/Hufflepuff/Ravenclaw values (insert traits)." So yes. I can be both tolerant and brave, for example. But if I value my tolerance above my courage- it means I am more of a Hufflepuff. Or, at least, this is how I see the things and this is how I would sort the Hat Stalls.
What makes a Hufflepuff stand as a house is not a problem of traits, I believe; it is a problem of how one sees the world. Our society is in desperate need of more Hufflepuffs.
Also, I'd like to point that Gryffindor is trying to steal some of Huff's traits. For example, you may hear that Gryffindors are just. I'm sorry; being just is a trait which defines Hufflepuff and only Hufflepuff. Many of people on the Internet forget about the "just and fair-play" story and those traits definitely wouldn't define a weak person.
A Hufflepuff is that one person who won't care about your background or about your well-known relatives: they care about you, as the person who stands in front of them. A Hufflepuff will never wish for fame because they understood that what matters is inside their hearts. As J.K. Rowling said, Hufflepuffs don't do things in order to show off or in order to be reckless, they do things for a different reason.
Most of the Hufflepuffs are people who have a little bit or a little more from all the other traits. Someone like Tonks. Wasn't she brave? Wasn't she smart? Yes. But she knew and most of the Hufflepuffs know that those traits don't necessarily have a connection with your heart, instead, it's hard to be a puff without being motivated by strong feelings and beliefs.
Hufflepuffs know how to see the beautiful in life, know how to smile on a rainy day and how to make you feel better. They are warm and kind.
And what's the problem with the "I'll teach the lot and treat them just the same." (which btw is such a Hufflepuff thing to do)?! Yes, indeed, not all of us have outstanding qualities so Helga seemed to value a certain struggle for being a better person. Not being a better person because of your achievements. Not because of your adventures. Not because of your brain. Because of your heart, because of your warmness, because of what's in there. Because the world is beautiful. And because you can try to make it better. That's the essence of Hufflepuff house.