Harriet's inlaid box is mentioned—count 'em—four separate times in the novel. Let's refresh our memories on when this box appears: In the opening scene, Harriet lets Lilia borrow her inlaid box to hold handkerchiefs and gloves. Yup. In Edwardian society you got a separate box for the items of clothing that are so important we don't even use them anymore. These are "keep your hands clean" gloves, not "keep your hands from getting frostbite" gloves, btw.
After Lilia disgraces the Herritons by marrying an Italian, Mrs. Herriton writes to her, kindly asking (i.e. ordering) Lilia to return the box that was "lent – but not given" to her (3.8). Halfway through the novel, at Gino's house, Miss Abbott notices the inlaid box covered in "a deposit of heavy white dust, which was only to be blown off one moment to thicken on another" (7.3).
And finally at the end after the tragic accident that killed the baby, Harriet goes crazy and starts muttering to herself about the inlaid box, repeating that it had been "lent—not given" (9.1). Because nothing says, "I'm devastated about murdering a baby" like getting your panties in a bunch about the ownership of a silly box.
Forster doesn't spend a whole lot of time describing the inlaid box, but the fact that the Herritons keep bringing it up shows how obsessed they are about it. They keep emphasizing that the box doesn't belong to Lilia, that it's theirs. Harriet is constantly reminding us how she had only loaned the box to Lilia, emphasizing just how possessive, petty and materialistic she is.
But the symbolic duty of the box doesn't stop with Harriet being a rhymes-with-witch. It's also symbolic of Edwardian society and its many, many rules. Rule #1: use gloves to keep your hands from getting dirty and/or suntanned, because the lower classes have dirty/suntanned hands. Rule # 2: use handkerchiefs to clean dirt off your face (to prevent yourself from looking lower class) and to dab away tears (because showing emotion is a big no-no in Edwardian society). Rules #3: keep both of these fairly useless items in a special box, because you can afford to. This inlaid box is the Edwardian equivalent of having an engraved, leather smartphone case: it screams "Look how rich I am!"
Also, it's a box. As in "boxed in." As in compartmentalization, and being trapped. Which is exactly what the Herritons try to do to Lilia: compartmentalize her as a no-good hussy, and trap her into their stifling Sawston life.