Over the next two months at the glorious John Rylands University Library in Manchester - built between 1890 and 1899 as a memorial to the greatest of the Victorian cotton magnates - visitors can see a small but beautifully formed exhibition about one of the city's best-known daughters. It has the unfortunate title "Elizabeth Gaskell: a Connected Life", which makes the novelist sound like a keen user of Twitter or Facebook, but you do see what the curators mean. It was not only that Gaskell had so many well-regarded literary friends (after the publication of her first novel, Mary Barton, in 1848 made her an overnight sensation, she joined a whole new social circle, one that included both Charles Dickens and Charlotte Brontë, whose biography she later wrote). Her Unitarian faith and social conscience also unfailingly tied her to those who were so often invisible to women of her class: the poor, the outcast, the criminal. The thought occurs that were David Cameron looking for ways to tighten up the flabby concept he calls "the big society" - and one sincerely hopes that he is - he could do worse than start by going straight back to the life of Gaskell, if not to her novels.

Thanks largely to Cranford, a slight book about the redoubtable ladies of pre-Wag Knutsford, Gaskell has long been a popular novelist, if not exactly a fashionable one. Her Victorian peers, once they had discovered the sex of the author of Mary Barton (it was first published anonymously), praised her mostly for her diffidence, her modesty and her ability to move her readers to tears - in other words, her considerable intellectual achievements were always veiled by "feminine accomplishment". This was how she continued to be seen until the 1950s and, perhaps, beyond.

Even in the early 1990s - the peak of the passion among English departments in universities for literary theory - when Gaskell was reassessed by feminist critics, she was somehow found wanting. They were kind, and righteously indignant on her behalf. But they mostly preferred the weight of George Eliot or the weirdness of the Brontës, or to show off by competitively digging up ever more obscure "forgotten" women writers whose names were excitingly unfamiliar and whose novels would look good packaged as Virago paperbacks.

Happily, "Elizabeth Gaskell: a Connected Life" makes no mention of all this. It takes her place in the canon as read and concentrates instead on biographical matters. If you have read any of the excellent biographies - I can vouch for Jenny Uglow's Elizabeth Gaskell: a Habit of Stories - you will know the tale: the beloved aunt who brought Gaskell up after the death of her mother in 1811, when Elizabeth was just a year old; the adored brother, John, a sailor who travelled to India to begin a new life and was never seen again; the marriage in 1832 to Reverend William Gaskell, a minister at the Cross Street Unitarian Chapel in Manchester; the publication of Mary Barton, whose shockingly realistic portrayal of workers living eight-to-a-room in damp, windowless basements infuriated the city's mill owners; the subsequent fame and fan letters (her new admirers included George Eliot, Charles Kingsley, Elizabeth Barrett Browning and Dickens, who later serialised both Cranford and North and South in his magazine Household Words). The John Rylands version of the tale, however, is punctuated with jewels from the library's remarkable collection of manuscripts and papers, as well as items borrowed from elsewhere.

Why are these writerly relics so touching and tender? Partly, it's the miracle of their survival. There is a cross-written letter (sentences scribbled over sentences, the better to save paper) from John that particularly pierces the heart. Like Mrs Hale in North and South, who stores her son Frederick's "yellow, sea-stained letters, with the peculiar fragrance which ocean letters have", Gaskell must have treasured letters from her brother. You have only to imagine its tortuous journey from India or Burma to see, in your mind's eye, the novelist pressing a piece of flimsy paper tight to her heart.

But it is also the awareness that, for Gaskell and her contemporaries, the written word was precious: not flung out in texts and emails, but always considered and wisely spent, as if it were money. And the sheer industriousness that writing then involved! A folio from the manuscript of The Life of Charlotte Brontë is on display and the hand all but aches at the sight of it - the dense script, the crossings out, every phrase having demanded a fresh visit to the inkwell.

Gaskell was a devoted wife and mother; only in the last third of the 20th century did people stop referring to her as "Mrs Gaskell". But she was also, strikingly, an independent woman - something this small show is at pains to emphasise, for all that it includes her Wedgwood teapot (rather covetable, as it happens). She relished the opportunity to publish work in magazines, sometimes making money on the same stories twice when they were later collected for publication in one volume. She loved to travel and the library owns her passport, a tiny, flimsy thing of which, one assumes, she was rather proud.

Most significantly, she maintained long correspondences, not only with other women writers, but with their male counterparts, too. (One of the most important of these - she wrote to him all her life - was with the American scholar and critic Charles Eliot Norton, a fellow Unitarian whom she first met in Rome in 1857; he had smiled up at the balcony on which she was standing.) Such relationships must have been marvellously reinforcing, particularly when controversy blew up. Her second published novel, Ruth, which addressed the question of the "fallen woman" sympathetically, prompted outrage among her husband's congregation, some members of which burned copies of it. But then her "hero" Kingsley - she loved him as much for his Christian socialism as for his writing - wrote to her praising it and all was well again, at least with her.

For me, though, the star item in the exhibition is a letter from a woman. In 1859, George Eliot wrote to Gaskell, informing her that she had "some affinity with the feeling which . . . inspired Cranford". Gaskell was thrilled. I stared for a long time at this letter - its spidery traces so generous, so full of fellow feeling - and felt very removed indeed from the newspaper I had read on the train up, full of snarky reviews of not-very-good novels.
Gaskell died suddenly of a heart attack in 1865. She was buried in Knutsford. William continued his good works in Manchester, where his two unmarried daughters, Meta and Julia, kept house for him. Here the exhibition ends, more or less (there is a small final section on her continued popularity; apparently there is a very active Gaskell Society in Japan). Afterwards, I headed out into the street. It was a beautiful, early-autumn day in Manchester, bright and crisp, and the city felt even more full of energy than usual. I decided to walk to the station by Cross Street. The chapel William Gaskell knew, a dissenters' meeting house erected in 1694, was destroyed by a bomb in 1940; Manchester's Unitarians now worship in a room at the base of an ugly 1980s office block, where they are surrounded by chain stores and pedestrian precincts - though they keep their end up by hanging colourful banners in the building's windows ("Lesbian and Gay Unitarians", read one, which made me smile).

It was a painful sight, this conjunction of the old and the new, of the godly and the godless. I couldn't find it cheery as a metaphor for our society and how its priorities have changed, for all that I'm not a believer myself. I suppose the chapel is, at least, a still point in a part of the city crammed mostly with people eager to part as quickly as possible with the contents of their wallets. I wondered what Gaskell would have made of it. I told myself that she would have found the good in it - though she would not have been restrained when it came to pointing out the bad - and walked on.

“Elizabeth Gaskell: a Connected Life" is at the John Rylands University Library, Manchester, until 28 November.