Nigel Mills (Amber Valley) won the private members' bill ballot, thereby securing first pick when choosing a Friday for his proposed law to begin its journey through the parliamentary process.
Conservative MPs secured the majority of key spots, while Labour's Ms Brabin came ninth, with Don Valley's Caroline Flint 17th and Bradford's Naz Shah 18th.
Ms Brabin told The Yorkshire Post: “I’m delighted to have had the 'luck of the draw', and I'm really excited to have my first Private Members Bill.
"Exactly when I can introduce the bill will be determined in the coming weeks but I need to have my idea soon.
"I’ve not decided on the subject yet, but what I do know is I want it be something that will make a difference in Batley and Spen and across Yorkshire – so please get in touch soon with any ideas!”
Mr Mills finished ahead of Tory colleague John Stevenson (Carlisle), Labour's Anneliese Dodds (Oxford East) in third and Conservative Anne Marie Morris (Newton Abbot) in fourth.
Labour's Lisa Forbes (Peterborough), Conservative former minister James Brokenshire and Liberal Democrat former leader Sir Vince Cable completed the top seven, who have the best chance of making progress with their draft legislation.
Deputy Speaker Sir Lindsay Hoyle said 395 MPs put their name forward to be in the ballot, with 20 drawn out for the different slots.
But there are usually 13 Friday sittings per parliamentary session, which makes it tricky for an MP to ensure their Bill completes all stages in the Commons before heading to the Lords.
Uncertainty over how long the new parliamentary session will last also means those successful in the ballot would see their hopes dashed if a snap election is agreed.
Sir Lindsay led the ballot with gusto. MPs signed their names against numbers, and numbered balls were then drawn out of a container.Introducing the clerk assistant helping with the draw, Sir Lindsay began: "We've got Sarah's shaker this year - shake them up!"
"All the fours, 44... all the threes, 33," he announced in bingo caller style as Labour's Preet Kaur Gill (Birmingham Edgbaston) and Conservative Damian Hinds (East Hampshire) were drawn.
"Boris's den, number 10," Sir Lindsay quipped, before announcing Labour MP Anna Turley (Redcar).
"174, Damien Moore... lucky number seven, Tracy Brabin," he continued.
Announcing first place in the ballot, he concluded: "And the winner of the 2019 ballot is... tension, drum roll - 209, is Nigel Mills!"