Of those three cities the OP mentioned, all of them had been seriously damaged by the wartime bombing and various battles associated with them. While reconstruction efforts were already well underway by 1947/48, they still had a long way to go. The debate for a capital for a future German capital for the forming FRG generally circles between two competing candidates- Berlin and Frankfurt am Main- and minor candidates like Kassel and Stuttgart. Bonn emerged late in the game as a dark horse candidate ( edit:- see discussion by /u/chakadal below and my qualification of "dark horse") and Adenauer tipped the scales in favor of Bonn.
There is a longstanding rumor that Adenauer pushed for Bonn because it was close to his residence in Rhöndorf and the Bonn would be a short commute for him while Frankfurt am Main would necessitate him to relocate. This somewhat petty reason for Bonn is somewhat implausible, but not entirely impossible. More importantly, the decision for Bonn shows the emerging strength of Adenauer. The Bonn decision was a personal play for Adenauer and the nascent CDU to put the temporary capital in a location they wanted. The temporary nature of the capital was why the location could become a political football. Both the CDU/CSU and the SPD held that Berlin would be the future capital of Germany once a final peace treaty had been hammered out between the German government and the four Allies. Thus the political parties could take a stand on the capital issue without fearing the wider repercussions of the decision. The rival SPD party had pushed for Frankfurt and the city itself had renovated the destroyed Paulskirche with an eye towards making it the future site of a German diet. But Frankfurt had connections with both the failures of the 1848 liberals and the Protestant German political tradition. As headquarters for the American occupation, it was also overrun with Amis, and this further made Frankfurt a less than desirable choice for German cultural conservatives. On a more practical matter, it also meant that the American military government had claimed some of the best (ie surviving and functional) real estate in the city such as the IG Farben building. Turning Frankfurt into a functional German capital would require expensive reconstruction, expansion, and repurposing of damaged structures.
Bonn, in contrast, was a sleepy German town that was Catholic, non-industrial, and had no real historical baggage. The war had only slightly damaged the city and it had no major Allied military government presence. Bonn was also much more congruent with the ideals of the CDU's Abendland and this ideological construct's associations with the West. As the birthplace of Beethoven, the city possessed capital as Germany's contribution to pan-European culture and it historically had been a go-between region for German Central Europe and France. Adenauer would state during the debates over the capital that Bonn was closer to the Western tradition and Western neighbors than Frankfurt, which was more connected to Prussia and traditions of German inwardness. Bonn's provincial qualities were non-threatening and hewed to stereotypes of a quaint "third Germany" that had no interest in either Prussian militarism or Hitler's aggression.
Moreover, and in keeping with the rumor about Adenauer's commute, victory over Bonn was a victory for Adenauer and, by extension, the CDU/CSU over the SPD. This was an important moral victory that gave the CDU/CSU a degree of initiative in the more contentious days ahead. This was why Hesse's CDU delegates voted against their Land's interests in having Frankfurt as the capital. By having Adeanuer publicly win this debate, the CDU/CSU further burnished his image as postwar Germany's inevitable national leader. The provincial nature of Bonn also signaled that the CDU/CSU had not given up on reunification and Berlin as the eventual capital. Bonn looked temporary in the late 1940s- there famously was only one main street. Such an image might not have been sustainable had it backed either Stuttgart or Kassel which were much larger.
Of course, turning Bonn into a federal capital was easier said than done. While the city's existing infrastructure suited a temporary arrangement, the increasing permanence of this temporary capital proved to be quite taxing to the nascent government in its first decades. Expanding the city to meet federal bureaucracy was an expensive undertaking and governmental architecture tried to create a "democratic" style that would eschew monumental celebrations of the state. The result was a somewhat bland architectural style that was unassuming, but also drab and not aesthetically appealing to many Germans of the postwar period. The machinations behind the scenes vote for Bonn was also somewhat suspicious. The FRG press reported on charges of bribery and other extralegal methods Adenauer used to gain his victory. This was one of several political scandals that appeared in the early years of the FRG that subsequent memory of the Wirtschaftswunder has eclipsed. As with much of the FRG, the Bonn decision was one in which a placid surface often concealed quite deep and tumultuous currents.
Sources
Feinstein, Margarete Myers. State Symbols: The Quest for Legitimacy in the Federal Republic of Germany and the German Democratic Republic, 1949-1959. Boston: Brill Academic Publishers, 2001.
Heidenheimer, Arnold J. Adenauer and the CDU: The Rise of the Leader and the Integration of the Party. 1977.
Irving, R. E. M. Adenauer. London: Longman, 2002.
Nielsen, Philipp. "Building Bonn: Democracy and the architecture of humility." History of Emotions-Insights into Research 2014, no. January (2014).