To understand the transition, we must first try to figure out why two-syllable given names were rare prior to and during Three Kingdoms. I will attempt to look at it more closely from right before Three Kingdoms, which is the two Han dynasties.

According to a study ( 单双名问题考察 ), in Records of the Three Kingdoms , 99% of names mentioned were of one syllable. In Book of the Later Han for the Eastern Han dynasty, it's at 98%. In Book of Han for the Western Han dynasty, it's at 77%.

While we need to take this with a pinch of salt as names cited in historic records were mostly famous people and those of higher status, which is not representative of the whole wider population , the dominance still clearly points to a preference of one-syllable names.

There were 2 main reasons why one-syllable name was preferred.

• Naming taboo. Known as 避讳, this has become very crucial and important since the rise of Confucianism. In summary, when one 避讳, one has to change his name to make sure that the character(s) used are not the same as that person they are trying to 避讳 (eg. the emperor) Comparatively, one-syllable names are easier to do so because well, others only have to change one word to 避讳. Hence two-syllable names are said to be " 为其难讳也 "《十三经注疏·春秋公羊传注疏 》, i.e. hard to 避讳. As such, one-syllable name was preferred.
• Wang Mang. When he was in power, he proposed that " 令中国不得有二名 " 《汉书·匈奴传下》. I.e. China should not have (people with) two-syllable names. He supported his claim with "讥二名，二名非礼也 " from the same source as the above point 《十三经注疏·春秋公羊传注疏》, which is a Confucian classic. Technically, there were various interpretations of this quote, but Mang simply took the interpretation that fit his proposal more, which is that two-syllable names were indecent, impolite. He also acted on his proposal and ordered the change of his eldest grandson from a one-syllable name back to his original two-syllable name, as a shaming and punishment for his despicable attempt to take over the throne. This is cited here

宗本名会宗，以制作去二名，今复名会宗。贬厥爵，改厥号，赐諡为功崇缪伯，以诸伯之礼葬于故同谷城郡。《 汉书·王莽传下》

Because of this incident, no one (at least amongst the upper class) dared to use two-syllable names when Mang was in power/the emperor. This also has lasting influences into the Eastern Han dynasty that followed, as nobody wanted to be associated with a traitor / someone who was publicly shamed. This could also be why we see the proportion of one-syllable names rose from 77% to 98% in Eastern Han as per what I mentioned earlier.

Now onto why a specific rise during Jin /4th Century

• The reign of Jin dynasty was one that's short lived and turbulent. In the 4th Century, Invasion and rebellion of the Five Barbarians occurred. This influx and invasion of nomadic 'barbarians' brought havoc and destruction to the middle kingdom, and along with that, displacement or destruction of the original Chinese culture with introduction of their own culture. This was cited to be unprecedented destruction of the traditional ritual and cultural system

宪章礼乐，寂灭无闻 《隋书·牛弘传》

This cultural 'disaster' brought 2 changes. First, people were no longer so fixated on certain traditional aspects, in this case, the preference for one-syllable names. Second, the introduction of nomadic people and their culture also brought along many new two-syllable names, as many of these people chose to retain their original, two-syllable names.

The combined effects of these 2 changes brought about more acceptance of two-syllable names and more appearance in historical records (as the nomadic people were very active in this period).

• Religious influences. During the Jin, a group of people known as the Seven Sages of the Bamboo Grove came to be popular and highly-regarded for their somewhat eccentric lifestyle and writings. These people were Taoists and disliked the Sima Clan (ruling family of Jin) and their rules. They preached "越名教而任自然" , essentially promoting a lifestyle away from the hypocritical 'etiquette' system of Confucianism, to simply be free and equal in all things in life. This somewhat Taoist concept, as well as the introduction of Buddhism , cause people to start using what is called '虚字' (virtual words) in their names to convey some sort of religious, spiritual meanings. Some of the more common ones are 之、道、昙、僧、灵 , all of them with some sort of religious implications.

Wang clan of Langya

Wang Xizhi and his sons are a good example, all of them used 之

These 'virtual words' also has the benefit of not being restricted by the aforementioned naming taboo.

晋王羲之子知名者五人：曰玄之、凝之、徽之、操之、献之。徽之子桢之，献之嗣子静之。祖孙父子，皆以‘之’为名，不以为嫌也。宋王弘子僧达，孙僧亮、僧衍，从子僧详、僧绛、僧虔，从孙僧祐，叔侄皆以‘僧’为名，不以为嫌也。 《 史讳举例 》

As these words serve no practical function and are just religious 'decorations' to one's names, they are not controlled by the rules of naming taboo.

As time past, people also began to realize that this additional 'virtual word' can also be a way for them to express well-wishes and blessings. For example

初，豁闻苻坚国中有谣云：‘谁谓尔坚石打碎。’有子二十人，皆以‘石’为名以应之。唯石虔、石秀、石民、石生、石绥、石康知名*《晋书·桓彝传附子豁传》*

The Jin general Huan Huo, after hearing the rumor that the enemy, Fu Jian's kingdom can only be destroyed by stones, promptly named all of his sons with the 'virtual word' 石 (stone) to fit this rumor and express his desire to defeat his enemy. From then on, people began to slowly adopt more words that are not so religious into their names to form two-syllable names with embedded meanings of well-wishes and ideals.

Hence, due to the changes brought about by religious influences and cultural shifts, from here on, two-syllable names became popular.

On a side note, as the population of China rose later on, there emerged more reasons and needs for two-syllable names , particularly during Tang and Song dynasties. But because they are not around the 4th Century period, they are not within the scope here.

Sources

《单名与双名：汉晋南方人名的变迁及其意义》 魏斌

《单双名问题考察》张书岩

《唐人姓名研究》 闫廷亮