Most discussions of which Supreme Court judge to appoint often revolve around the position that the proposed indivdual is likely to take on a set of 'hot button' topics - abortion, 'right to die', the death penalty etc. There are also some euphemisms for this such as asking about the 'judicial philosophy' of the proposed candidate. In fact, judges don't seem to entirely willing to meet in full any expectations that may be engendered as regards on the basis of their apparent 'judicial philosophy'. For example, Rehnquist's theoretical position on the Miranda rules on the rights of suspects weren't entirely reflected in his decision not to overrule that precedent in Dickerson.
This type of reticence is probably explained by the fact that judges can often perceive that there is a 'wider picure' that sometimes mitigates against a single -minded pursuit of one's own views even where the law does indicate scope for discretion. For example, in Dickerson, Reinquist was apparently swayed by wider arguments of legal certainty. Such considerations suggest that there is a danger of over-emphasizing the political aspects of these appointments (the fact is that the incumbent president will hold most of the best cards so far as this dimension is concerned). Maybe a more naive approach is preferable in which proposed candidates' legal credentials (in terms of their general nous and ability to take a wider view of a problem) are genuinely seen as the key aspect of the appointment so far as public scrutiny of that process is concerned.