His fame as an investigator and teacher, and as a man of brilliant and powerful parts, was now being widely spread. Pupils came to him, not only from various parts of England, but from America, Australia and Japan. At the York Meeting of the British Association for the Advancement of Science, in August, 1881, he was chosen as one of the General Secretaries. In April, 1881, the honorary degree of LL.D. was conferred upon him by the University of Glasgow, and in November of the same year the Royal Society gave him one of the Royal Medals in recognition of his embryological discoveries, and at the same time placed him on its Council.
At Cambridge he was chosen, in the autumn of 1880, President of the Philosophical Society, and in the December of that year a brilliant company were gathered together at the Annual Dinner to do honour to their new young President. Otherwise nothing as yet had been done for him in his own University in the way of recognition of his abilities and services; and he still remained a Lecturer of Trinity College, giving lectures in a University building. An effort had been made by some of his friends to urge the University to take some step in this direction; but it was thought at that time impossible to do anything. In 1881 a great loss fell upon the sister University of Oxford in the death of Prof. George Rolleston; and soon after very vigorous efforts were made to induce Balfour to become a candidate for the vacant chair. The prospect was in many ways a tempting one, and Balfour seeing no very clear way in the future for him at his own University, was at times inclined to offer himself, but eventually he decided to remain at Cambridge. Hardly had this temptation if we may so call it been overcome when a still greater one presented itself. Through the lamented death of Sir Wyville Thomson in the winter of 1881-2, the chair of Natural History at Edinburgh, perhaps the richest and most conspicuous biological chair in the United Kingdom, became vacant. The post was in many ways one which Balfour would have liked to hold. The teaching duties were it is true laborious, but they had in the past been compressed into a short time, occupying only the summer session and leaving the rest of the year free, and it seemed probable that this arrangement might be continued with him. The large emolument would also have been grateful to him inasmuch as he would have felt able to devote the whole of it to scientific ends; and the nearness to Whittinghame, his native place and brother's home, added to the attractions; but what tempted him most was the position which it would have given him, and the opportunities it would have afforded, with the rich marine Fauna of the north-eastern coast close at hand, to develop a large school of Animal Morphology. The existing Professors at Edinburgh were most desirous that he should join them, and made every effort to induce him to come. On the part of the Crown, in whose hands the appointment lay, not only were distinct offers made to him, but he was repeatedly pressed to accept the post. Nor was it until after a considerable struggle that he finally refused, his love for his own University in the end overcoming the many inducements to leave; he elected to stay where he was, trusting to the future opening up for him some suitable position. In this decision he was undoubtedly influenced by the consideration that Cambridge, besides being the centre of his old friendships, had become as it were a second home for his own family. By the appointment of Lord Rayleigh to the chair of Experimental Physics his sister Lady Rayleigh had become a resident, his sister Mrs Sidgwick had lived there now for some years, and his brother Gerald generally spent the summer there; their presence made Cambridge doubly dear to him.