At the moment, the head of a household fills in the details of those living in a property but ministers want people to register individually to combat fraud.
Those refusing could be issued with a "parking-style" fine by local councils.
Labour said fines could only be justified if they were an incentive to register and they must be "reasonable”.
Speaking in a debate on the issue, Mrs Laing - who has been pressing for this change for many years and has raised the matter in debates in the House of Commons many times - said the Bill which contains the proposed changes is “will improve the electoral register’s comprehensiveness and accuracy".
Mrs Laing said: "It is long overdue. It is absurd that, in the 21st century, a person’s right to vote depends on the head of household filling in a form”.
She added: “Each individual member of our society should be responsible for registering themselves to vote and should have the vote that they deserve.
“I have never understood why the Labour party has been so reluctant for the last two Parliaments to go ahead with that obvious modernisation of our electoral administration system.”
Mrs Laing added: “My greatest concern is the idea that young people will not register to vote if their mother or father does not fill in the form for them. What absolute nonsense!
“I shall go further: if a young person cannot organise the filling in of a form that registers them to vote, they do not deserve the right to vote.
“If someone is responsible enough to exercise their right to vote to decide the Government of this country, or at any level of local government, they should be responsible enough to register to vote.”