Matlab plots can display data-tips but these cannot be interactively moved except to the 4 corners of the data point.

This utility enables the user to interactively drag any newly-created data-tip, anywhere in the Matlab figure. A dashed line connects the dragged data-tip with the original data point.

The new draggable functionality only affects new data-tips, so it can be turned on/off to enable standard and new data-tips to co-exist in the same plot (see screenshot).

Syntax: draggableDataTips(newState)

Examples:
draggableDataTips('on') or draggableDataTips(true) turns the new functionality ON for new data-tips.
draggableDataTips('off') or draggableDataTips(false) turns the new functionality OFF for new data tips.

Warning:
Relies on undocumented functionality.
This utility fails in HG2 (R2014b+) - please don't bug me about it, unless you found a solution...
On HG2 (R2014b+), use Allen's submission: https://www.mathworks.com/matlabcentral/fileexchange/69799-dragdatatip

Technical description:
http://UndocumentedMatlab.com/blog/draggable-plot-data-tips