There's actually a lot to this expression, including a lot of nuance and some understanding of each character's personality. Bear in mind, I have a pretty pedestrian understanding of the full nuances of the Japanese language, but I'll be happy to cite my sources where need be.
First, let's start with Yuru Yuri. Akari is made out to be the decoy protagonist very early on, and as such, the fact that she's has very little presence is a running gag in the series. In the first episode, around 16:40 is everyone comes to that realization (spearheaded by Kyouko).
Around this time, the nickname actually comes up, and this is the beginning of her being teased with "Akari~n!" (and turning invisible). Notice that the first episode's intro doesn't do anything fishy with her or her visibility (in the same way that the subsequent intros do).
Next, let's move on to normal Japanese naming convention. In Japan, it's only considered okay to call someone by their first name if you're really close to them. From what I've seen, nicknames are slightly more relaxed, so you'll see and hear people calling each other by their nicknames quite often in anime. From this, it's implied that Yui, Kyouko, and Chinatsu are at least incredibly friendly with Akari enough to call her by any sort of nickname.
Adding ん to the end of a name (and it depends on the name) invokes a cutesy, childish sounding variant of it, and can be considered to be a cutesy sort of nickname for the person.