Memory of water
The memory of water is a phrase mostly associated with homeopathy and the controversial research of Jacques Benveniste and his colleagues. This team reported that particular solutes subjected to sequential physical processing and dilution appeared to have some biological effects that were different from the "control" effects of the water used for the dilutions.
Water is not simply a collection of molecules of H2O, it contains several molecular species including ortho and para water molecules, and water molecules with different isotopic compositions such as HDO and H218O. These water molecules as part of weakly-bound but partially-covalently linked molecular clusters containing one, two, three or four hydrogen bonds, and hydrogen ion and hydroxide ion species. In addition, there are always adventitious solutes in liquid water. Even double-distilled and deionized water always contains significant and variable trace amounts of contaminating ions, and different samples will differ in the contaminants that they contain.
There is some support for the notion that water can have properties that depend on how it has previously been processed (that is, water has, in some sense, a kind of "memory"). The experimental evidence indicates that the "memory" are due primarily to solute and surface changes occurring during this processing. In particular, water, as a result of repeated vigorous shaking, might include Redox molecules produced from water, dissolved atmospheric gases and airborne contaminants, Silicates - tiny glass "chips", nanobubbles and their material surfaces, dissolved ions, including from the glassware. It may be contaminated by material that adheres to the surfaces of glassware, for example by bacterial material. There might also be some effects of successive shaking on water structure - "clustering" of water molecules.
These mechanisms are not mechanisms of memory in any cognitiove sense; the term memory here is used as a metaphor, implying only that the past history has a discernible influence on the present properties.