Durable modifies the minimum number of hit points you can recover from rolling a Hit Die
To see how the Durable feat works, it is important to look at all the relevant text for the given mechanics. (Something I am sometimes guilty of failing to do, as well.)
First, start with regaining hit points through expending Hit Dice.
Short Rest PHB, p. 186
For each Hit Die spent in this way, the player
rolls the die and adds the character’s Constitution modifier to it.
The character regains hit points equal to the total (minimum 0).
Note the (minimum 0) in parentheses at the end, which I bolded. Now read the text of the Durable feat.
Durable PHB, p. 166
When you roll a Hit Die to regain hit points, the
minimum number of hit points you regain from the roll equals twice
your Constitution modifier (minimum of 2).
It seems clear to me that the feat modifies that minimum.
The question and other answers are all focusing on a debate over whether the term "the roll" refers to the number on the die verses the number on the die plus the constitution modifier. I feel that that has been debated to death, and the answers from ipsec and Adeptus cover the relevant arguments well.
It all strikes me as a bit of tunnel vision, which is easily resolved by looking at the wider context: namely, the minimum hit points regained is what is being changed.
As such, for the example presented of a +4 CON modifier with a d12 Hit Die, the minimum is simply 8. This is an increase from the defined minimum of 0, and the practical minimum of 5 (from rolling a 1 and adding 4.)
Who benefits from the Durable Feat?
As this feat modifies the minimum hit points regained, it benefits those characters the most that either had a minimum of 0 or 1 without the feat (so, CON modifier of 0 or less), or those with a high constitution modifier.
Note especially that a character with a constitution modifier of +1 would gain no benefit, as their practical minimum was already 2 (1 from the die roll and +1 from their CON modifier), and their new minimum is still just 2.
There is also a question that directly addresses the benefit of the Durable feat, for anyone that wants a more complete analysis and nice graphs to look at.