A person’s name is a way of distinguishing him/her from others, of identifying the individual as distinct from anyone else. As such, a name must have distinctive features so that thereby we are able to, with few exceptions, identify one individual as separate from another. It is thus important that within the mechanism of a name we are able to narrow down, to a great extent, the individual to whom we are referring.
Within the Jewish world, this is accomplished, generally, through the use of an individual’s personal name with a further reference to a parent. In most cases, the parent mentioned is the father so a full Hebrew name would generally be one’s individual personal name with the further mention that the individual is the son or daughter of the father. When the father is not Jewish, however, this reference to the father is not applied within Jewish Law. The essence of your question is as such: what other criteria is used in the name identification of a person if the father’s name is not used, such as in a case when the father is not Jewish? (The same question may be asked when, for example, the father’s identity is not known.)
The usual custom in such cases is to follow the view of Rema, Shulchan Aruch, Orach Chaim 139:3 who states that the name of the father of the mother (i.e. the maternal grandfather) should be used instead of the father’s name. There are those, however, who disagree. See Mishneh Brura 139:10. The second option which then seems to have the most weight is to simply refer to the individual as the son or daughter of Avraham Avinu, Avraham our forefather. So in answer to your question, you have two basic options: either to refer to yourself as the son/daughter of your maternal grandfather or as the son/daughter of Avraham.
I should mention, perhaps, that there are actually also two other possibilities. As reference in a name is given to the mother in certain cases -- such as in prayers for health – the option to refer to you as the son/daughter of your mother is also presented, within the literature, as a possibility. The idea that perhaps you should just refer to yourself with one name, your personal individual one, is also found. These two latter options, however, reflect limited, minority opinions and, in the determination of your Hebrew name, it would be best if you choose one of the first two options.