The duke is an extremely proud man, and he wanted his wife—his last duchess—to love him the best, to pay him the most attention, and to think of his gifts to her as the most important and wonderful ones she could ever receive. Instead, however, "she liked whate'er / She looked on," he says. Any small gift would make her blush with pleasure, and she bestowed her smiles on anyone who offered her any gift at all, whether it was a "bough of cherries" a servant brought her from the orchard, a pretty white mule, or her husband's "gift of a nine-hundred-years-old name." She failed to thank him more than she thanked anyone else, and she failed to appreciate his incredibly valuable gift more than anyone else's.
Her generosity of spirit and gratitude to anyone who showed her a kindness are not the qualities the duke wanted in a wife. He wanted her to love him and his gifts the best! And she did not pander to his sense of pride, did not stroke his ego, and did not make him feel important. The line that references his name shows the pride he takes in his pedigree and family history, and it shows that he values these things much more than he values love or even human life. As a result of his insecurity and wounded pride, he "gave commands" which stopped "all smiles [...] together." He seems to have had her killed so that he could try again and find another wife who would treat him and his "nine-hundred-years-old name" with greater reverence.