The answer lies within the intricacies of the British peerage system.
In 1884, Tennyson became 1st Baron Tennyson, having declined a baronetcy twice before, in 1865 and 1868. He was the first man in history to be given a title for his services to poetry.
As a baron, his official title was Baron Tennyson or Lord Tennyson. If you use his first name, his correct title is Alfred, Lord Tennyson. Lord Alfred Tennyson is incorrect; those words in that order denote the courtesy title of a younger son of a duke or marquess.
So, why did Tennyson continue to use his first name after he became a baron?
This is a good question, as it was not necessarily typical for a baron to be known by his first name. An example of this is George, 6th Baron Byron, who you probably know better as Lord Byron.
While we can never know for sure what Tennyson’s reasoning was, we can make educated guesses. By 1884, Tennyson’s best-selling and most famous works—such as In Memoriam (1850), Maud (1855), and Enoch Arden (1862)—had already been published under the name Alfred Tennyson. All of his fame and achievements were linked to this name. It would, no doubt, have been confusing if, at such a late stage in his career, he had suddenly started styling himself as Lord Tennyson.
Another reason may have been that he was sentimental about his old name. Tennyson was reluctant to accept the peerage at first and turned it down repeatedly. He even asked whether the title could be granted to his son, Hallam, when he died, without him ever using it. On another occasion, he turned it down by telling Gladstone that he and Emily preferred to “remain plain Mr and Mrs”. Upon his acceptance of the title, Hallam records his father as saying he would “regret” his “simple name” for the rest of his life. Perhaps signing himself Alfred, Lord Tennyson helped him to feel closer to the Alfred Tennyson he had once been.