I was wondering what a moral system that says things are good if they benefit me would be called. At first I was thinking of normative egoism, but it seems like normative egoism says that everyone should act in their own self-interest, rather than just my interests. So my question is is it the case that there is another system already in philosophical literature that describes this view perfectly, is there not one and this is a completely new position, or am I just misunderstanding normative egoism?
The ethical position you outline is a logical possibility but I am not sure that it has been held - or at least held widely enough to attract a label. That's not a point against you since you ask what such a position would be called or (in effect) appropriately could be called. If there isn't a label, can a suitable label be invented?
A position without a name
I don't think there is any standard name or label for the position you outline. (There is after all no apriori reason why such a name or label should exist.) This is no problem logically if the position can be coherently described, which it can be. We then face merely the task of devising a suitable label. In strictness this would be a label by which the positiom could (or better: could reasonably and illuminatingly) be characterise. Since there no existing label, how can be know what the positio would actually be called?
The position is plainly a form of ethical egoism but, equally plainly, to say this is not enough since it does not distinguish your form of ethical egoism from others. We need a label for the differentia.
Types of ethical egoism
Ethical egoism - mark 1
'A person, X, ought, all things considered, to do an action if and only if that action is (at least judged by X to be) in his or her overall interests'.
Ethical egoism - mark 2
'A person ought, all things considered, to do an action if and only if that action is in the general interest (where each person's interests are coordinate with every other person's interests, and each person is regarded as an 'end in himself'.
Without coordination of interests there cannot be a general interest.
Ethical egoism - mark 3
'A person ought, that is to say that every person ought, all things considered, to do an action if and only if that action is (at least judged by every person to be) in my, X's, overall interests.'
'Mark 3' is, I take it, with whatever refinements or qualifications you might want to add, the ethical position you want to label. So if this is the position, what is the label?
I hesitate to endorse 'ethical solipsism' because of 'solipsism's' indelible association with the idea that only one person exists. In the moral world you describe there is a plurality of persons. (The Wiki entry for 'ethical solipsism' is only a stub and does not so far as I know register any widely accepted usage.)
However, the same consideration applies to any other label one might come up with because the fact is - I think it is a fact - that the ethical position you outline is a logical possibility without an actual name.
The interesting reference above to mysticism has one consideration against it in my view. Mystical perspectives deny the ultimate separateness or distinctness of persons but the sort of ethical egoism with which you are concerned stresses separateness: it stresses the existence of a plurality of persons, separate from X but owing to X the obligation to do serve exclusively X's interests. Clever and perceptive though the reference to mysticism is, I don't think it carries the right associations.
Then what label are we to use? One we have invented for the purpose, I think. 'Ethical monism' is a possibility if it conveys the idea that all moral action should converge on serving the interests of one identifiable person - X in the example. But 'ethical monism' might mean so many other things that I can't favour this suggestion.
'Ethical singularism' is my best suggestion, inadequate no doubt:
'A person ought, that is to say that every person ought, all things considered, to do an action if and only if that action is (at least judged by every person to be) in my, a single person X's, overall interests.'
I have revised some formulations in Tibor R. Machan's 'Recent Work in Ethical Egoism', American Philosophical Quarterly, Vol. 16, No. 1 (Jan., 1979), pp. 1-15. None of my suggested labels is his.