Why when we add an additional month to the year do we only add it in Adar? Why not have a second Cheshvon or a second Shevat, etc.?
We add the additional month in order to keep Pesach in the spring, as the Torah mandates (Deut. 16:1). So it is added as close to Nissan as possible, in order to make explicit the link between the extra month and its purpose.
Tosafos (Sanhedrin 11b, s.v. אין מעברין) says that since we have verses in Tanach (Esther 3:7 passim) that call Adar the "twelfth month," then necessarily no other month can be inserted before it, otherwise some years it would be the thirteenth.
The Mechilta (Bo 2) cites a different reason: the year should be similar to the month in the way it's extended - in both cases, the extension (a day or a month) comes at the end.
Sanhedrin 11b: Before the setting of the calendar, those in charge of being meaber would base their decision on two of three factors which were related to the ripeness of the produce during adar (is the barley ripe enough to call Nissan "Aviv", will the fruits be ripe for Shavuos), and could not be determined earlier.
The pamphlet "הדף היומי בהלכה", published by Dirshu, issue 4, Ⅰ Adar 5774, cites K'dushas Levi (Ki Sisa) and S'fas Emes (likutim for Adar) as saying that the month of Adar (whose zodiac sign is fish) corresponds to Yosef (which, like fish, is unaffected by ayin hara) — see the Tur (OC 417), who says each month corresponds to a tribe. Thus, it had to be the month to split into two, since Yosef is the tribe that split into two, M'nashe and Efrayim.
Adar is the last month in the year. So from a simple technical standpoint it's the easiest to add a month then. Especially if you want all the other months' numbers to remain like they should be (and more importantly - compatible with what the Torah calls them)