I think they were called pigs because of the sound they make - sort of like hedgehogs.
The problem is that they are exotic, which means expensive. In the European food paradigm, you don't eat exotic pets (we don't do roast parrot, either, though the Romans did). So the key step would be to make them unexotic. I also think they don't deal well with low temperatures, but surely it can't be a big problem given they live in the Andes.
So, by way of an idea: Some improvement-minded European rules in the eighteenth century decides to advocate raising guinea pigs as a solution to suburban meat supply (IOTL, rabbits filled that niche). IIRC, keeping and breeding guinea pigs is quite similar to rabbit-raising, except that they take better to being caged. By the 1850s, a guineas pig cage is a common sight in the back garden or kitchen of every upper working class household, and many agricultural labourers keep them for variety alongside the more economical chickens. Initially, the recipes used are those for hare and rabbit, but later on, we see specific guinea pig preparations. large varieties bred for meat are also likely to replace the cute, fluffy kind we're used to. Rabbit becomes a rural food.