Has anyone ever named a war after their own country or faction?

Well, I am not sure if it counts for you, but the Russians named their struggle against Germany (WW2) as "Great Patriotic War". Stalin used it on his radio message on 3. July 1941

see: http://www.1000dokumente.de/index.html?c=dokument_ru&dokument=0029_stj&object=translation&st=&l=de

The Falklands War between the UK and Argentina in 1982.

Three of the British-French wars are known in America as King William's War, Queen Anne's War and King George's War, which were the American sections of the War of the League of Augsburg, War of the Spanish Succession and War of the Austrian Succession. Since there already was a King George's war, the American section of the Seven Year's war had to be named after something other than the King.

During the period of the Republic of China (this period is not limited to the reign of Chiang Kai-shek, this Republic refers to the period between the fall of the Qing Dynasty and the Communist Party came to power in 1949) wars (except with Japan and the communists, for the communists battle, the government at the time called the "repression of bandits" during the period before the Japanese invasion.) were basically named after factions, such as the first, second, Three Zhili-Fengtian wars which is start by two warlord named Zhang Zuolin and Duan Qirui. First_Zhili Fengtian_War, Second_Zhili Fengtian_War, and the Anti-Fengtian_War.

two sheds already mentioned the Napoleonic Wars, which do kinda qualify(they're named after the leader of the country), AND

The French Revolutionary wars(guerres de la Révolution française in French) are a perfect example for what you're asking. They're named after the... we can say faction or political entity, that caused and won most of them, and are known under this name both in France and abroad.

Pink's War, a campaign fought between British India and the Mahsud in 1925. The war was initiated by the British, but it was named after the British officer in command of the campaign.

Surely an obvious example is the American Revolution? And likewise for many other revolutions.

Winston Churchill, before fighting it: "The battle of Britain is about to begin. Upon this battle depends the survival of Christian civilisation."

At least two other answers have mentioned that the French refer to the Napoleonic wars by the name of their leader, but miss that they are also referred to (in English) by the faction the British were in. The War of the First Coalition, The War of the Second Coalition, all the way through The War of the Seventh Coalition.

One example that comes to my mind is the War of the Triple Alliance, perhaps more commonly known as the Paraguayan War, which was won by the said Triple Alliance (Brazil, Argentina and Uruguay).
EDIT – But to be honest, I don't actually know nor could I find out whether it was called like that while it was being fought; it's pretty probable that it was called either "Paraguayan War" by the Allies and "War against the Triple Alliance" by the Paraguayans at the time.

Hmm... Perhaps the War of the League of Augsburg / War of the Grand Alliance / Nine Years' War would count, at least with respect to the first two of those names. According to Wikipedia,

The Grand Alliance was a European coalition, consisting (at various times) of Austria, Bavaria, Brandenburg, the Dutch Republic, England, the Holy Roman Empire, Ireland, the Palatinate of the Rhine, Portugal, Savoy, Saxony, Scotland, Spain and Sweden. The organization, which was founded in 1686 as the League of Augsburg, was known as the "Grand Alliance" after England and Scotland joined the League (in 1689). It was originally formed in an attempt to halt Louis XIV of France's expansionist policies.

Unfortunately, in a quick search I can't seem to find anything that says authoritatively whether contemporaries used either of those two names, though they have been used quite a bit since then.

Naming a war after the leader of our side (especially if he wants to be remembered for the victory, even anticipated) like in "Napoleonic Wars" as referenced by two sheds seems to be the most natural case of naming the war after one's side.

In Clone Wars it's a different case: naming the war after a key or new weapon. I don't know about any war named like this, but I know of one battle: the battle in which Antioch I. defeated Galatians is known as the "Elephant Battle". I see no reason why a war couldn't be named after a weapon/technology in a similar manner.

One potential answer is "The People's Crusade." This certainly refers to the people fighting it (peasants instead of noblemen). I haven't found any primary source material for contemporaries calling it "The People's Crusade", but this source seems to suggest that it was called "The Popular Crusade" which is fairly close.

As two sheds and Steve Jessop pointed out, the word crusade comes from the Latin for cross which refers to the Christians who did the fighting, so all of the crusades were actually named by one side to refer to themselves and that terminology was used at the time.

I can think of an example of this from the ancient period:

The Lamian War(323–322 BC): was known to the ancient Greeks as the "Hellenic War".

Obviously there were many wars in ancient Greece that we could call "Hellenic Wars" but this particular one was explicitly noted by Diodorus Siculus as such. Independent Greek states fought on both sides of this war: Athens and the Aetolian league on one side, Macedonians and Boeotians on the other side.

The Austro-Prussian War is currently known in Germany as "Deutscher Krieg", or "The German War" - though it was originally known as "Preußisch-Deutscher Krieg", or "Prussian-German War".

Another contender are the Napoleonic Wars--or the Guerres napoléoniennes, as they are called in France.