Nick Hurd, the son of former foreign secretary Douglas Hurd, was the first of the 20 MPs whose name was picked.
Private Member's Bills allow MPs to introduce legislation on a subject of their choice. About six are debated in the House, but very few become law.
Mr Hurd and others near the top of the list have not yet decided what their bills will be about.
Traditionally those at the top of the list are lobbied hard by people who want specific issues to be chosen.
Eleven Conservatives and nine Labour MPs were on the list. No Liberal Democrats or members of minor parties were among the 20 names drawn out of the total 471 MPs taking part.
The successful 20, in order, are:
Nick Hurd
Tim Yeo
Caroline Spelman
Gary Streeter
Graham Stringer
Robert Walter
Sir John Butterfill
Paul Farrelly
Martin Caton
Richard Ottaway
Roger Godsiff
Shailesh Vara
Alan Duncan
John Hayes
Barry Sheerman
John McDonnell
Sarah McCarthy-Fry
Michael Meacher
Emily Thornberry
Francis Maude
In previous years private member's bills have included a ban on fox hunting, most recently introduced by Labour backbencher Gordon Prentice in May 2000. It ran out of time, but the ban was eventually introduced in a government bill.
Successes were a law banning shops opening on Christmas Day, introduced by Kevan Jones, and one restricting the sale and use of fireworks, introduced by Bill Tynan in 2003.
In 2002 support was given to measures to tackle overgrown hedges. Although time ran out for it to become law, the government adopted it as a policy and pushed it through.
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