The words "Never Knows Best" comes from Mamimi's cigarettes.

Interviewer: "What does 'Never Knows Best' mean? The English on her [Mamimi's] cigarette."
Tsurumaki: "Oh, that? Originally, it was a cigarette on a postcard I have."
Interviewer: "What did it say on the cigarette?"
Tsurumaki: "It was a close-up picture of a cigarette that had 'Join London' on it. And it was probably
handwritten, not printed. I thought it looked cool and that I would mimic it. It's not a
cigarette brand. There's no such brand in Japan."
Interviewer: "I thought that it might be the title of a song."
Tsurumaki: "Oh, no. I'm not sure if it has a real meaning or not."
So the director Tsurumaki Kazuya says that he's not sure if the phrase "Never Knows Best" has any "real meaning or not." The series is already complicated and wild enough, I guess the director was just trying to put our minds at rest by saying that he's not "sure." That's bull! He's the director, of course he knows what it means! He's just keeping it a secret. And secrets don't make friends. Let's figure it out!

Now, from my solid six year experience as a gas station cashier and the thousands of cigarette packs that I have sold from the 20-30 different brands (each with at least 10 different flavors and types) that are in stock, I have yet to sell a cigarette that has a message inscribed on the actual cigarette paper. The only kind of messages I have ever seen are on the actual box which usually says "20 Class A Cigarettes" or "Surgeon General's Warning: Smoking By Pregnant Women May Result In Fetal Injury, Premature Birth, And Low Birth Weight."
The usual messages that are on the cigarette boxes are very important, whether they be the type that someone prefers such as "Light 100's" or the required "Surgeon General's Warning" that are on each box. I use to think that the message that Mamimi wrote was nothing important, but come to think of it, it could be a special thought or message that is very sacred and dear to her. Let's investigate!
Now, I'm not familiar with the term "Mother never knows best," but I have heard of "Mother knows best." With the "never" in the phrase, it seems to make the message come off with a feeling of angst and emotional negativity, almost with an insincere attitude about it. The same applies to the other two choices of "Father" and "Tasuku" making it seem much, much more personal. Now with Takkun, it comes off as just a regular comment - no "real" meaning.
In my opinion, with "Mother," it makes it seem as if a betrayal happened, sort of like an abandoned feeling. Maybe Mamimi's mother wasn't happy with her school work and told her to pack her things and leave. With "Father," it makes me of think of pedophilia (extreme sexual children loving fetish). Maybe her father's abusive nature or her mother's abandonment led her to live a life of carefree solitude (such as smoking at a young age, dropping out of school, living on her own). With "Tasuku" as the noun, it makes me think of disappointment and regret - perhaps Tasuku, Mamimi's ex-boyfriend who went to America, never really took a sincere liking or interest in Mamimi, hence him not knowing her that well. Again, these are mere speculations. But with Takkun (Naota, the black kitty, the robot-dog thing), it's just a regular comment that describes what's going on. For example, a cat doesn't really "knows best" by almost getting killed in the M.M. robot battle in episode two and the robot-dog thing is pretty clueless too by being force fed due to Mamimi's fuel for vengence. Naota doesn't seem to "know" Mamimi that well neither and only serves as sort of a sexual-relief outlet for her (and probably himself). Although he "cares" the most for her out of all the other characters, it seems that he only does it out of curiosity (such as asking questions all the time) and sexual enjoyment. Even in episode five, Mamimi herself knows that Naota is only caring for her for the physical favors he'll get in return (next to the Blue Cafe, he tries to give her a really wet kiss) as oppose to getting to "know" her with more-emotional, less-physical aspects. But who really knows?
Ignoring that there isn't a noun, what does "Never Knows Best" seem to mean? Well, it suggests that as people, we don't always know the best path to choose in a given situation. That seems like a simple and honest message that affects everybody everyday.

Mamimi's purpose in burning the cigarettes, more specifically cigarettes written with the phrase "Never Knows Best," may serve as both religious and therapeutic outlets for her. For example, on Mamimi's cigarette incense, she purposefully did not put a subject before the phrase "Never Knows Best" as seen clearly in the screenshot above. Perhaps, the ritual that she is performing absolves and relinquishes everyone from the meaning of the message (hence the lack of a preceding noun) - sort of like a spiritual cleansing. Or, by not including a specific subject, Mamimi could have actually emphasized her message to include everyone kind of like a teacher commanding her whole classroom without actually spe. Thus, her ritual with the incense can now be interpreted as a sort of universal truth that either blesses or damns everyone.
The second incident is more obvious. In the second episode, Mamimi is doing her little Kanti ritual with the inscribed cigarettes all lit around her and also with grill lighters attached to her head (weird...). Right as she is about to go kiss Kanti, Naota's head explodes again with a Medical Mechanica robot.
In the sixth episode, Mamimi is just chillin' under the bridge next to the river again. She calmly smokes her cigarette and then the terminal core a.k.a. robot dog thing crawls out from the river and befriends her. She blows the smoke into it and continually feeds it until it's the size of a school bus and a cement truck put together.
Now I understand that Mamimi's cigarettes didn't summon out Ninamori's M.M. robot nor did she even smoke when the grand-daddy of all M.M. robots in episode five came out of Naota's head, but could it be possible that some of the robots respond to her inscribed cigarette's smoke? Although possible, I think it's highly unlikely and merely coincidental.
Now, pretending that her inscribed cigarettes actually did cause the M.M. robots to come out and attack, what correlation do the two have? The only relationship that I see between "Never Knows Best" and "Medical Mechanica" is... nothing. But, "Never Knows Best" did beat "Medical Mechanica" by nearly 72,000 results on Google-Fight! :D

She still believes in fairy tales (definitely not wrong with that; I still open closet doors hoping to find my way into Narnia) and acts like a child. Heck, she's still exploring her sexuality like what other children do when they hit puberty. Mamimi can't even separate fantasy from reality when she plays her game FireStarter.

So, maybe, it's really "Mamimi never knows best." It could simply mean that she doesn't really know anything, or at least, she doesn't know the right course to take. She is a confused girl and really does not have any real set goals in her life. Possibly, after she became a photographer and travelled on her own at the end of the series, I think she may have left her cigarettes behind... She now "knows" and understands what she wants in life I guess. She wants freedom, creativity, and photography.