My sense is that two Obama transition officials--Michael Froman and, to a lesser extent, Sonal Shah--played important roles. Froman, a friend of Obama's from law school, started off as an economic aide in the Clinton White House, then became a deputy assistant secretary on Treasury's international side around the same time Geithner was promoted to a similar position. He also served as then-Treasury secretary Bob Rubin's chief of staff between 1997 and 1999, at which point Geithner was promoted to assistant secretary and then under secretary. For her part, Shah was a special assistant to two colleagues of Geithner's at Treasury and ended up working with him closely.
I'd guess they favored Geithner at Treasury for two reasons:
1.) Both Froman and Shah were personally very fond of him, according to former colleagues. Not that this was unusual at Treasury. Geithner was very well-liked by co-workers generally, especially so among those lower on the organizational chart. As I wrote in my recent piece:
At meetings with subordinates, he'd rarely sit at the head of the table. In fact, he'd rarely sit at all, preferring to pace around the room prompting people for input. When he briefed a higher-up, Geithner's habit was to bring along the bureaucrat who'd worked with him on the issue. ...
Within the building, Geithner became legendary for his self-deprecating humor. Shortly after taking over at Treasury in 1995, Rubin convened a meeting with his senior officials and asked them to introduce themselves, prompting an arms race of resume embellishment. Finally, it was Geithner's turn. "Well, I've mostly been in high school," he cracked. Everyone doubled over in laughter.
2.) My hunch is Froman and Shah were keen to see a new face at Treasury. Froman is roughly Geithner's age; Shah is five-to-ten years younger. While former colleagues say both of them liked and respected Summers, he was always several rungs above them at Treasury. It would hardly be surprising if they wanted to see someone closer to their own cohort get the top job.
That's not to say these were the only factors in play. Or that Froman and Shah were the lone, or even most important, decision-makers in the Geithner appointment. (Geithner obviously had a lot to do with it himself; I'm sure other transition officials were involved as well.) But I doubt it hurt his chances either.