Apparently, Cardano had learned a formula for solving cubic equations from Tartaglia, who had sworn him to secrecy, and in any event, not to publish it without giving Tartaglia due credit.
Cardano published it under his own name, claiming to have learned it elsewhere from a man named Ferro.
"Everyone" knows these facts. So why do people keep referring to it as Cardano's formula, rather than Tartagia's formula, or even Ferro's formula?
In short, because only positive numbers could be used as coefficients at the time there were several cases of the cubic that had to be treated separately. Before Cardano only depressed cubics (with one of the powers missing) were solved. He introduced the substitution that reduces general cubic to a depressed one, and made formulas for depressed ones public for the first time, with his own derivations. Ironically, the formula under his name is not due to him, but he was the first one to "publish" it, while acknowledging del Ferro's and Tartaglia's contributions. Since the use of complex numbers was rudimentary at the time even Cardano's generalization did not resolve the case of a cubic with three real roots, the irreducible case, as Bombelli pointed out. It was later resolved by Viète, who related it to the angle trisection problem.
Cardano's masterpiece, Ars Magna (1545), was the most comprehensive treatise of algebra until Viète's Isagoge (1591). Unlike del Ferro and Tartaglia who dealt with special cases only, Cardano provided systematic treatment based on substitutions of all thirteen cases of depressed cubics, and of the general case. It is no accident that it was his student, Ferrari, who solved the quartic following the same methods (his solution was also included into Ars Magna). Moreover, Cardano acknowledged the possibility of negative ("fictitious") roots, and in one example considered complex ("sophistic") roots for the first time. Ars Magna's influence is often compared to that of Copernicus' De revolutionibus, and it did a lot to promote new methods in algebra at the time. Most mathematicians learned the cubic solution formula from it, which explains the naming. This is similar to "Pascal's" triangle. It was known even in Europe long before Pascal (curiously, Tartaglia claimed it as his), but Pascal was the first to systematically study its properties.
The discovery itself is a fascinating story. It was Scipione del Ferro who solved the first depressed cubic. At the time it was common for mathematicians to call each other to "duels", where having a secret trick for solving equations no one else could solve was a big advantage. These duels often carried monetary prizes, and affected recognition and appointments in ways publications do today. So del Ferro kept it a secret, which he entrusted to his student Fiore. Tartaglia challenged Fiore and shortly before the contest solved del Ferro's case and one other (he admitted though that what helped him solve it was knowing that solution existed). Needless to say he won the duel.
After futile attempts to solve cubics by himself Cardano convinced Tartaglia to give him the formula with a promise that he would keep it a secret until Tartaglia publishes it in a book he was writing at the time. However, Tartaglia took his time and eventually Cardano found out that del Ferro discovered the formula before Tartaglia. He no longer felt bound by the promise, and published the formula in Ars Magna. Tartaglia was understandably livid. The coda was Ferrari challenging Tartaglia to a duel after discovering a solution to quartics. That one Tartaglia lost.
It is not "attributed to him", because as you mention yourself, everyone knows the story. It is only "named" after him. Very many things are named with other names than their inventors, or discoverers or first persons who proved a them.
Some random widely known examples are: Schwarz Lemma (he did not state it in full generality), Schwarz inequality (due to Cauchy and Bunyakovski), Schwarzian derivative (considered by Lagrange), Alexandrov compactification, Riemann zeta-function, and many many other things.
Name of a theorem or formula is just a label. It has has to be a) well recognizable, and b) commonly accepted. Who proved it first, is not so important (except for the author:-) and in many cases several people contributed.
"America" is also not named after Columbus (or whoever "discovered" it).
EDIT. Let me also mention the well-known "Arnold's Principle": If a thing is called after a person, this indicates that this particular person had nothing to do with it. Sir Michael Berry replied on this with a Theorem: Arnold's Principle applies to itself.
One thing to take into account is that recognition of the discoveries goes to the first who publishes it. You may discover the secret of live, the universe and everything, but if you just hide the knowledge you cannot claim its prior discovery when someone other publishes them.
For example, Gauss did investigate the possibilities of non Euclidian geometry, but chose not to publish his findings. So, hyperbolic geometry is also known as Lobachevskian and not as Gaussian geometry.
So in this case, even if Cardano's disclosure was dishonest, the naming would just follow the usual conventions.
