Which Paris arrondissment to stay in?

If you were only staying a few days, I agree with the most common advice given here, which is that the 4th, 5th, and 6th arrondissements are most convenient, since you can walk to more attractions. But with a full month, and a tight budget, closeness to the Seine is less important, and you'll get more bang for your buck in the double-digit arrondissements. You'll also see more Parisians and fewer tourists.

So, in your situation, I'd worry less about the arrondissement, and more about the particulars of the apartment (I assume you will use one, instead of a hotel, for a stay of that length). Does it have enough space for everyone to be comfortable for a month? Is there a washer? A dryer? Does it have an elevator? If not, how many flights up is it? Remember that the first floor is above the ground floor, so a "third floor" apartment is up three flights of stairs. If you're going in the summer, is it air conditioned? Are there noise issues? Is it near a Metro station served by more than one line, which means less changing? Etc, etc.

You will find nice and not nice apartments in all arrondissements. Each arrondissement, in turn, has different kinds of neighborhoods. For example, people often use "5th arrondissement" and "Latin Quarter" interchangeably, but the Latin Quarter is actually only a small part of the 5th. However, there are supermarkets, parks, cafes, etc in all arrondissements.

Harold is right and he explained it well. Generally, the higher # the arrondissement, the more budget friendly it will be.