That was this family's specialty: strange conversations.
— Tomoya to Mei Sunohara, regarding the Furukawa household following a conversation about who Nagisa would allow to check for a nonexistent scar on her butt, Clannad
Now remember, as far as anyone knows, we're a perfectly normal family...
An ensemble of bizarre characters who are related, or might as well be. Unlike the Dysfunctional Family, we as the audience plainly see the family is extremely well adjusted, supportive and loving — more so than some "real" families. They are also easily able to absorb friends, acquaintences, and distant relatives into their structure.
This is coupled with a range of quirks easily labeled "bizarre" by any of their peers.
It's also very convenient for heroes to have these, as they're not bound by the Masquerade, weirdness is normal.
Likely part of the natural growth of a Pretty Freeloaders group. For a larger grouping, see Quirky Town.
The Hiiragizawa household in Cardcaptor Sakura. The "dad" looks like a ten-year-old kid, and his "children" are a genderless magical being posing as a teenage schoolgirl and a winged cat that looks like a toy most of the time.
Although with only two members, they don't make as dramatic a showing as some examples listed above, arguably the Koiwais from Yotsuba&!. Certainly, in his way Dad is nearly as quirky as Yotsuba, if more laid-back about it.
In Unstrung Heroes: young Steven Lidz' home, to a lesser extent. Especially quirky, however: the labyrinthine apartment (filled with hoarded junk) in which he lives with his eccentric uncles for awhile, after his mother's illness exacerbates the awkwardness of Steven's relationship with his father. Franz—actually Steven, now rechristened by his uncles; It Makes Sense in Context—undergoes an disorienting yet often enjoyable identity crisis. Then, as it turns out, at least one of his uncles is a bit more than merely eccentric. Franz must navigate between his uncles' pride in their heritage (complicated by imagined anti-Semitism lurking everywhere), and his inventor father's 100% materialist (i.e. anti-spiritual) worldview.
The Moomin family, from the childrens' book series by Tove Jansson.
The Weasleys fit this, due to their seven children (nearly all with extremely different personalities), Mama Bear mother, eccentric father, weird pets (hyperactive owl; ancient, enfeebled owl; and ghoul), all crammed into a small, ramshackle house and, oh yeah, they're all wizards.
If they qualify, then they're normal compared to the Lovegood Household.
Subverted by The Young Ones, in which none of the characters in the house can actually stand one another for any significant length of time. This doesn't count as an instance of Dysfunctional Family, as the characters in a Dysfunctional Family show are permitted to get along with one another despite their differences.
Dan: I also liked that reasoning because it gave me the idea for this comic, and I love writing scenes like this. There's a certain madness to the Dunkel household that, in my opinion, makes the Verres household look relatively sane. It's enough that I feel I must now assure you, the audience, that there are no questionable ingredients in Mrs. Dunkel's brownies.
The Planet Express staff from Futurama straddle the border between the Quirky Household and True Companions. In the episode "Future Stock", Fry even says, "We're not a traditional family, like the Johnsons next door or the lesbian coven across the street, but we're still a family!"
The boarders in Hey Arnold! are portrayed this way in several of the later episodes.
The Chan family, in Jackie Chan Adventures, have a rather strong sense of this, consisting of a twelve year old who can sneak in anywhere with almost ninja-like efficiency (whether she's wanted or not), a grumpy, snarky old witchdoctor, a former sumo wrestler turned villianous Tank, turned Gentle Giant chi wizard in training and a rather stressed out archeologist with a knowledge of martial arts as good as... well, he's Jackie Chan, you do the math.
The Flyn-Fletcher Family from Phineas and Ferb. A father whose an expert on random and obscure antiques, a mother who was once a one-hit-wonder, a semi-neurotic teenage girl who likes screaming at cheese, a pet platypus whose secretly a special agent, and two brothers who do everything.
The Cake household in My Little Pony: Friendship Is Magic. For one thing, PinkiePie is part of it. There's also Mr. and Mrs. Cake's precocious foals Pound Cake (a pegasus) and Pumpkin Cake (a unicorn), and Pinkie's toothless pet alligator Gummy.
The Goof family on Goof Troop is a single parent household where the father is a clumsy idiot (Goofy) and the son is a snarky High School Hustler (Max). The closest thing they have to problems are Max feeling embarrassed when people say he reminds them of his dad, but they are extremely loving and close, especially compared to their neighbors.