I suspect, or fear, this practice may turn out to be the core of real Christianity, the main means of salvation, in the future - if the mainstream Christian Churches continue their decline and corruption, and assuming the Eastern Orthodox Church does not have a resurgence in the West.
I heard of the Jesus Prayer first, many years ago, in Franny and Zooey by JD Salinger - and I would recommend reading the book discussed there - The Way of a Pilgrim, in the translation by R.M French.
This also is wise and helpful: http://www.philokalia.org/on_the_prayer_of_jesus.htm
But the Jesus prayer was originated in Orthodox Christian societies, where a core knowledge of Christianity could be taken for granted - and that is certainly not the case now: ignorance and profound misunderstanding are the current norm; and erroneous or false teaching much easier to find than truth!
So (contra what is said in Franny and Zooey, and what is *superficially* implied by the Way of a Pilgrim) I do not think the Prayer of Jesus would 'work' (or even begin to work) unless the person saying it knew the essential meaning of the words.
For this, some prior teaching is necessary - which in our era may have to come from reading.
My own case is so very atypical that I really have no idea what would be right for other people - but I found this essay extremely helpful - Theosis: the true purpose of human life by Archimandrite George: http://www.orthodoxinfo.com/general/theosis-english.pdf.
However, for me it came after several years of wide-ranging and mostly misguided reading, reflection and discussion - so it may not mean much to someone jumping straight-in.
The point is that Orthodoxy provides a very clear aim - Sainthood - and a (relatively) clear definition of what this entails (ascetic struggle) and what success brings (living in heaven upon earth).
Theosis is a path on which all can and should embark, and it is being on this path that matters to salvation - although almost all will fail to reach the end of the path and most will advance only a few baby steps - the necessary start being repentance (a turning-away from the Kingdom of Man towards the Kingdom of God) and by effort obtaining only a *foretaste* of heaven on earth. Ultimately, it is enough.