A) While claims that she can predict would itself work, the placement the of the modifier, from a five-minute palm inspection, makes this version seem to convey that her claim, rather than her prediction, is based on a five-minute palm inspection.

B) This one repeats the modifier placement issue present in A. Also, this version is missing a that before she has the ability. While some would argue that a that is not necessary there, from what I have seen the GMAT prefers that that be used after bridge verbs such as claim.

In any case, the modifier placement issue is solid decision point that one can use to eliminate this choice.

C) This is idiomatically correct and via the placement of the modifier correctly conveys how the five-minute inspection is related to her being able to predict future life experiences.

D) claims the ability of is just not idiomatically correct. Could you maybe correctly decide to eliminate this choice because of the meaning, arguing that she is not claiming the ability but rather claiming to be able? Maybe.

E) Similarly, claims being able to is not idiomatically correct. Alternatively, possibly this choice could be eliminated via analyzing meaning, in that she is not claiming being able, she is claiming to be able, but without knowing which is the idiomatically correct form one might have difficulty eliminating this choice with certainty.

In any case, because it conveys a clear meaning, is well structured and is idiomatically correct, choice C is the best answer.