In men the ratio between the length of the second and forth finger are thought to be an indication for the amount of prenatal testosterone. It allows for some inferences on personality traits that are associated with masculinity such as aggression.
There are a few reports of doctors identifying some specific brain tumors via handshakes.
White spots on fingernails can be a sign of a possible Zinc Deficiency.
Hand grip strength seems to be a predictor for lifespan.
Most of what's commonly done under the label of palmistry however doesn't seem to be backed by research.