This is probably not a question that has a single answer, since our hypothetical aristocrat in exile managing to successfully raise and army and challenge (presumably) whoever exiled him or her is a situation that already has great deal of moving parts. The reason the aristocrat was exiled factors in, as does the whoever exiled them, and how secure that group's hold on power was compared to remaining support for the exile, and any other groups that simply didn't want the usurpers hanging around. Basically, the situation you are positing is already a situation where the normal state of things are in fluctuation.

Really, I'm not even sure this happened as often as you might think. Napoleon is an example that has already been brought up as an example, but he spent less than a year in exile before returning to widespread acclaim and defections to his side. That's one way to do it, I suppose. I have another example though, which might show a slightly more complicated path from worst to first.

Tepanecs, Acolhua, and Mexica, oh my!

Just to briefly set the stage, in the late 14th/early 15th centuries CE, a Nahua group known as the Tepanecs were busily setting about conquering the Basin of Mexico. From their capital city, Azcapotzalco, the Tepanecs, led by the long-lived statesman Tezozomoc, had brought large swathes of the Basin under their dominion, including the rich southern agricultural lands. All that remained was the sparesly populated north and the eastern lake basin, the lands of another Nahua group, the Acolhua.

Around 1414-15, war between the Tepanecs and Acolhua broke out and, despite a strong showing by the Acolhua, the Tepanecs won out. The Acolhua cities were assigned as tributary vassals to the victorious Tepanecs and their allies. Notably, the city of Texcoco was assigned to the Mexica of Tenochtitlan (the people we think about when we think about Aztecs). More notably for your question though, many Acolhua elites fled their cities, including a young teenager named Nezahualcoyotl, son of the ruler of Texcoco. In the most dramatic tellings of the story, Nezahualcoyotl witnessed his father being struck down by assassins before his very eyes. Regardless of that bit of melodrama, he ended up fleeing to the neighboring Tlaxcalan lands with many other Acolhua elites, as a sort of prince-in-exile.

Tepanecs 2: Post-Tezozomoc Boogaloo

Shortly after securing the eastern lake shore, however, Tezozomoc finally passed away. A succession struggle then occurred, with the son he had chosen to succeed him being deposed and murdered by another son, Maxtla. Now, Tezozomoc had always shown favor to the Mexica, who had grown from starving refugees to a substantial economic and military force under the protection and dominion of Tezozomoc.

Maxtla, however, did not share that warmth and saw the Mexica as threats to his power. One of his early acts after deposing his brother was to have the ruler of Tenochtitlan and his young son assassinated. Naturally, most people do not take kindly to having their leaders murdered, so an uncle of the slain leader was elected to rule and sort of cold war began between the Mexica and the Tepanecs.

In the decade or so between the fall of the Acolhua and the murder of the Mexica monarch, however, Nezahualcoyotl had returned to the Basin of Mexico and settled in Tenochtitlan. This should not be surprising as the Mexica were the overlords of his city (and other Acolhua holdings), but also because the ruling families of Texcoco and Tenochtitlan had intermarried; Nezahualcoyotl's own mother was part of the ruling dynasty of Tenochtitlan. He had thus spent his late teens and early twenties living among the Mexica nobility, though without any real power of his own.

That's One Hungry Coyote

The aggressive action of Maxtla threw the politics of the region into turmoil. The Tepanecs technically controlled most of the Basin, but allegiances in Postclassic Mexico were... fluid (as the Mexica themselves would find out a century later). Both the Mexica and the Tepanecs were sending out embassies to the various groups in the Basin, trying to find out how everyone's loyalties would sort out.

In this sorting, many pro-Tepanec Acolhua nobles aligned against the Mexica, including relatives of Nezahualcoyotl still living in Texcoco. The timing is a little fuzzy here, but apparently either at this point, or even a little earlier, the Mexica recognized Nezahualcoyotl's claim to Texcoco and he assumed the role of Tlatoani (lit "Speaker," but "King" is a common translation) of that polity. Even though he had no real power -- his pro-Tepanec relatives still controlled the city -- the title gave him political clout.

As the official ruler of Texcoco, Nezahualcoyotl was able to reach out to the Acolhua still in exile in Tlaxcala, who could then reach out to their contacts still residing in the Basin, thus forming a web of Mexica support on the eastern lake shores. Brundage (1972) astutely notes that one selling point may have been that, if the Tepanec won, the Acolhua would still be a divided group of subordinate cities. If the Mexica won, there was at least the possibility that the Acolhua would gain some measure of independence as allies to the victors.

The Conclusion You Knew Was Coming

The problem with history is that all stories are already told. If you know anything about Aztec history, then you know that the Mexica won the war with the help of the Acolhua. Nezahualcoyotl drove out his Tepanec sympathizing relatives in 1433, and Texcoco became the preeminent city of the Acolhua and the second most important city in the Basin of Mexico behind Tenochtitlan. Nezahualcoyotl himself would reign for decades and is remembered as a great statesman, architect, judge, philosopher, warrior, and poet. So not bad for someone who was living as a noble without a city until the political equilibrim of the Basin of Mexico got flipped turned upside down and he ended sitting on his icpalli as the tlatoani of Texcoco.

So, how does one raise an army?

As for the technical details of what you're asking about, the key is organization. Cities in the Basin were divided into calpulli (wards/barrios). This system was developed and formalized during the Aztec Imperial Period, some sort of calpulli-like system may have existed in the Basin of Mexico for centuries, perhaps even back to Teotihuacan. The basic idea is that each calpulli would be responsible for neighborhood upkeep, policing, and schooling, which included military training.

When mobilization occurred, the call would go out the calpulli, and the available men would be organized into a squadron. This squad would then join up with other squads to form larger units, and so on and so forth. The point being that social structures for raising and equipping large numbers of men for war already existed in Mexico, and no doubt in other areas as well. What our hypothetical noble in exile needed then, was not some new innovative way to marshal troops, but the political, social, and legal authority to utilize those pre-existing systems. As with the example of Nezahualcoyotl, gaining that authority as someone who had previously been deposed from power might be dependent on the political landscape shifting so that those who exiled you are now vulnerable to a challenge, and the strength of your claim to wielding that authority now exceeds theirs.

Sources and Such

Brundage, BC 1972 A Rain of Darts: The Mexica Aztecs - Dated and overly dramatic, but does a great job of synthesizing primary sources.

Hassig, R 1988 Aztec Warfare: Expansion and Imperial Control - The authoritative book on the sundries of the Aztec military.

Lee, J. 2008 The Allure of Nezahualcoyotl: Pre-Hispanic History, Religion, and Nahua Poetics - A sort of anti-hagiography of Nezahualcoyotl and Texcoco. Makes its case a little too stridently at some points, in my opinion, but is still a treasure trove of information on the subject.

Alva Ixtlilxochitl, F (~1640) Historia de la Nación Chichimeca - The Acolhua side of history, written by a descendant of Nezahualcoyotl. No English translation of which I am aware.

Duran, D (~1581) *History of the Indies of New Spain - The Mexica side of history, written by a Spanish monk who grew up in Texcoco and Tenochtitlan. The 1994 Heyden translation is the standard.

Lets say you are the French king and you didn't like the English very much, and the former king of England (and his son) wind up in your French court. Would you execute him? Kick him out? Heck no! Pretenders to a throne (especially ones with really good claims like James and Bonnie Prince Charlie) are very useful resources to have.

If you deem the time is right (let's say because the English are going to oppose you on the continent, as they did repeatedly), you send a pretender off to Scotland or Ireland with some troops, and see if they can't take out or destabilize the English for you.

In regards to the communication issue, raising troops in France wouldn't be much of a problem if you are using Expats like the Irish Brigade (http://en.wikipedia.org/wiki/Irish_Brigade_(French)) or French troops, but it was absolutely a horrendous problem when communicating and coordinating with Scotland or Ireland, as the English tended to get in the way. This caused countless problems with timing issues and landings, as friendly Spanish and French troop ships over the years were blown off course, landed in the wrong place, failed to rendezvous, or were intercepted by the English.

The Jacobite Rising of 1715 suffered a massive communication problem - the Scots had already risen and been mostly defeated by the time the pretender landed, leaving him nothing to do but turn back around for France.

Geoffrey Parker's "The Army of Flanders and the Spanish Road" has very good illustrations of recruiting in the early modern era of the Eighty Years War (1566-1648).

The Spanish empire commissioned recruiting captains and contractors. The recruiting captains, with assistants, would travel from village to village, put up a small show by "veterans" telling tales of adventure, riches, glory, etc. Those who sign up are then given payment for signup, and then at some point they are all collected up to form one unit (squadron, or battalion, etc.) and then shipped off to their deployment zones.

In the early days of the Eighty Years War this worked well enough, however towards the end there was general reluctance to join the army and more critically shortage of able-bodied men. So recruiting captains started to get recruits from the prisons, offering freedom after their term of enlistment. Or, escape from a death sentence.

During the bad times in Castille, recruiters would offer food in exchange for enlistment.

Would anyone mind giving a more Roman-leaning answer? For example, in the Roman Civil Wars, various people go around raising private armies all the time, like Pompey did in the war with Sulla. I'm assuming he went to lands where his family was important and drew on the support of patrons?