Go to Players → Memberships and drag the membership cards into the order you want. The order saves automatically, and players see the same order in the app.
Reorder memberships
Go to Players → Memberships.
Drag a card up or down.
Wait for "Order saved".
Good to know
Dragging a membership from Public memberships to Private memberships hides it from players. Dragging it back shows it again.
If your club is part of a multi-club company, company memberships appear in the list too, greyed out with a Company badge. You can move them to set where they appear at your club. Players see club and company memberships in one list.
Moving a company membership to Private memberships hides it at your club only.