How does a series earn the Weekly Shonen Jump cover?

There is no contest page where a series wins the cover. The cover belongs to the series holding the issue lead colour chapter, the front of magazine feature slot, and both are assigned by the editorial department. What readers control is the evidence the editors read: every issue carries a survey postcard on which readers pick their three favourite chapters of the week, and those returns are the magazine famous popularity signal.

Eiichiro Oda explained the inside order in a One Piece SBS answer that the Library prints at face value: to be precise, the order is decided every week by the Jump sub chief editor; the most popular series are placed toward the front; and sometimes the editor decides the magazine is pushing a particular series that week. Popularity moves a series forward over time, colour pages are both a reward for popularity and a tool of promotion, and a new anime adaptation reliably earns a series a push. It is an edited magazine, not a chart.

So a cover means three things at once: the editors chose this series to sell this week issue, the survey record made that choice safe, and the chapter inside opens in colour. A debut chapter 1 almost always takes the cover, because the launch is the push; a final chapter almost never does, by house policy, which is why the one famous exception, Kochikame in 2016, is a story of its own.