I believe the correct answer is that you place that son in positions of great responsibilities and importance like governing provinces or leading armies. Ottoman succession has been called "open succession" because there is no preferential order to succession (i.e. primogeniture, oldest son). An Ottoman prince was educated in the Harem (yes it was not just a personal pleasure palace) and once he completed his education he would be given responsibilities. For example you would give your preferred son a province of great prestige and importance while also being close to the capital (Constantinople) so by the time of your death, he would be closest to the capital where he can rush and claim the throne while have greater access to resources than his brothers. Of course your son would need to help himself as well by being competent and gathering support in court, the military, religious authorities, or what ever channel he can find. Bigger army diplomacy was a practice widely used in succession crisis across the world, including Ottoman ones, so the prince with greatest following and reputation would have the advantage. Your sons may even come into conflict with one another during the later stages of your reign to position themselves for the throne. Factional rivalries would form purely on succession, so party A wants your son Mehmed to succeed you while B wants Selim. These factions included everyone from generals, court officials, to even your own wives and concubines. Sometimes this conflict would spill from courtly intrigue into all out war resulting quasi civil wars. A jealous son or one who could see the writing on the wall may even seek to overthrow you to circumvent competition. Selim I defeated his brother to succeed his father, and his son Suleiman I would execute 2 of his own children because of possible courtly intrigues and rebellions leaving Selim II as his heir. Finally, if your hypothetical son did succeed you, then he would probably have his male siblings executed (by strangling with silk) or locked away in the Harem on permanent arrest (unless he dies without an heir). In short you can try the rig the game by giving preferred son extra pieces, he would still have to play it.

Side question: Would there be any precedent in the Ottoman Empire for Sultan ExtraSoftHandker to choose a designated heir?

While MockOnVoltaire is correct, this only applies after the reign of Ahmed I. The Sultans often did interfere, and during the reign of Suleiman the Magnificent (1520-1566), the sultan killed his son Mustafa, who was considered the most suitable successor, and actively supported his son Selim to kill his other son Bayezid. This action, has been theorized as planting the seed of downfall of the Ottomans. The practice of fratricide, which is the killing of brothers, led to 10 great sultans, and with Suleiman, the quality of the sultans began a downward slide. The reason the practice was institutionalized in the first place was because the law dictated that the oldest living son was to be sultan, thus creating the incentive for brothers to kill each other. This practice was assumed to produce the most worthy sultan, for the one who survives, would be the smartest, bravest and most deserving to lead the expansionist Ottomans. However, in order to win, the brothers would lead armies against each other, which they would produce due to their positions of governors of various provinces. The Sultan would place his sons in various governorship in order to prepare them for the throne. So after the Sultans death, it would be a race to Istanbul/race to kill each other. So to answer your question, if the sultan wanted a particular son to the succeed, he would place him as governor of a province closer to Istanbul which was wealthier. As Suleiman the Magnificent grew old, he had two sons left alive, Selim and Bayezid, and it was Bayezid who drew the ire of the Sultan for his hesitation to accept the governorship of his executed brother Mustafa. This prompted the Sultan to support Selim to kill Bayezid, and Bayezid was killed in 1559 after a battle against Selim. This interference of the Sultan has led many historians to describe as one of the turning points of the rise of the Ottomans. While the practice of fratricide was cruel (particularly after the reign of Ahmed I, whose institutionalized the imprisonment and isolation of siblings produced mentally unstable sultans), it produced what was considered to be strong sultans, in the first half of the history of the Ottomans.

In the period you are talking about the best way to ensure succession to a favored son was to send them to govern a province near Istanbul and to ensure that the mother maintained a position of political power. You could also consider killing a strong heir, your son, whom you didn't like.

Being governor gave the prince a military force to pursue his claim. Being near the capital allowed the prince to capture the political center quickly, before other claimants.

Selim II, who succeeded Suleyman the Magnificent in 1566, was the first Sultan to succeed without having governed a province as a prince. The eldest son, Mustafa was executed by his father because he posed a, not improbably, threat. Bayezid, the only other brother died in 1561, leaving only one heir. Selim was the son of Suleyman's favorite concubine.Pierce, p. 80-83

After 1550, the practice of sending a son to govern provinces died out, as did fratricide. Succession (and reproduction) was managed by the Dowager Empress (valide sultan) and favored the eldest male in the family as a whole. Pierce, p. 93

For more detail, I can't recommend Leslie Pierce's The Imperial Harem enough. It is an amazing book in its detail and its revolutionary take on the historiography. Everything above is just taken from it.