This is a frequently used English expression which does not exclusively relate to diamonds but rather is used as a metaphor.
The expression is used to suggest that someone could be a real success as they have the basic qualities inside of them however that they lack the finishing touches. In general the expression is most commonly used in relation to people although it can also be used to describe inanimate objects.
There is a clear consensus that the expression relates to the fact that an unpolished diamond, whilst not perfect, holds a large amount of potential. This potential is often overlooked due to the initial appearance of the diamond which can look like a common pebble and thus the expression is used to describe those who hold great potential which has not yet been recognised.
The first recorded usage of the expression dates to the year 1624 in John Fletcher’s novel A Wife for a Month. The phrase “she is very honest, and will be as hard to cut as a rough diamond,” is used in order to describe the wife being described.
An example of the usage of this expression can be found in a quote by English journalist Daniel Defoe who stated that “the soul is placed in the body like a rough diamond, and must be polished, or the luster of it will never appear.”