Tilly's name is Matilda. It was typical of Mum and Dad that they wouldn't let her be called Tilly when she came to them because that was the unusual shortening of Matilda, the unconventional one, and you never did unconventional things, you never did things to draw attention to yourself. So they called her Matty. Of course, they hardly ever called her anything, but when they had to, it was Matty.
Source: wiktionary