In some countries, women have always kept their name. In other countries, married women must change their name. In New Brunswick, no law ever forced women to drop their name. But here as in most countries, it is custom, not law, which maintains the widespread tradition of taking the husband's name. The custom is relatively recent and, in Europe, was started by families wanting to establish family rights to land inheritance.
Some women feel strongly about wanting their children also carrying their name.
To them, it's like painting a masterpiece and not being able to sign it. When the mother has taken the husband's name, the children are inevitably given their father's name, and this usually means they will identify with that name to the exclusion of their mother's. In Spain, among other countries, a child's last name is the mother's name. There is now a Lucy Stone League and web site dedicated to 'Name Freedom'. It is hard to fathom that all Lucy Stone had to do to be remembered 150 years later was to say her name.