One important component of the liberal case for Elena Kagan seems to be that she could exert a strong influence on the Court's median vote, Anthony Kennedy. Here's Jeffrey Rosen:
Obama has signaled that he wants a justice who can win Justice Anthony Kennedy to the liberal side of the Court in 5-4 votes. Given Kagan’s demonstrated success winning over skeptical conservatives at every stage of her career, she seems ideally suited for this role.
And half-seriously, Jonathan Zasloff:
Kagan may not be a great scholar, but she is enormously skilled at impressing older colleagues — and that’s just what the doctor ordered for this appointment.
Essentially, any Supreme Court appointment this cycle has two tasks: 1) vote the right way; and 2) convince Anthony Kennedy to do the same. Kagan seems to have the skills to do that.
Is there any evidence whatsoever that Kennedy is susceptible to lobbying for votes, subtle or otherwise? A fairly large literature has emerged about the internal workings of the Rehnquist Court, and I've read a painfully high percentage of it, but I'm not aware of any documented case in which the influence of another justice has caused Kennedy to switch his views.
The most prominent case in which he switched his position after the conference vote -- the school prayer case Lee v. Weisman -- didn't seem to have anything to do with personal dynamics on the Court, and by all of the accounts I'm aware of William Brennan's attempts to influence Kennedy were a dismal failure. I'm happy to be corrected if anyone has an example, but I don't know of any actual evidence that brown-nosing can win Kennedy's vote. I think part of the problem -- which was also true of Sandra Day O'Connor -- is that some Court observers conflate moderation with indecisiveness. Just because a justice's votes are less predictable than some of their colleagues' doesn't mean that they are to be subject to manipulation.
This point isn't just about Kagan. I agree with Dahilia Lithwick that this also largely applies to Diane Wood, who has demonstrated some ability to influence conservative colleagues in a more relevant context. How interpersonal relationships will play out is nearly impossible to predict, and the effect in any case is likely to be trivial.