Question

What major arguments made by Quaker Abolitionists were to end the institution of slavery?

Answer 1

Their argument was very similar to what most people believe today about slavery. They were hard-core christians so they believed that everyone was equal. They thought slavery was unjust and immoral because of their beliefs. Not many other people thought this at the time, but eventually their beliefs came to be more common.

Answer 2

It's important, possibly, to clarify the phrase "hardcore Christians", since a number of sects of Christianity, perhaps the majority, centered themselves around a fearful, wrathful God (as He might have generally appeared throughout the Old Testament), rather than the all-compassionate one He is often shown to be in the New Testament.
Quakers centered their belief around a God of the latter sort, and were a sect of Christianity which, while being quite fundamentalist at times, deemed all of mankind to be of the same ilk and all capable of receiving the same grace.
Such a view almost directly opposes Calvinists (one of the other "more prominent" sects of Protestantism arising from Martin Luther's famous renouncement of the Catholic Church; Puritans being followers of the Calvinist 'model'), who believed, to some extent, that members of their sect possessed the only souls that would ever be given any redemption by God, and that the rest of the world was essentially doomed, no matter what quantity of quality prayer was undergone, no matter to what extent humanity repented and revoked sin.

Quakers were hardcore in terms of their beliefs, but their expression of faith differed in many ways from what might be the more "traditional" model of Christian celebration. They were less about decorated ceremony and idolatry and more about living the faith day in and day out.
... but I've digressed long enough.