The stone of good luck should grant its +1 luck bonus to the initiative check.
The PRD has the following to say about this item:
This small bit of agate grants its possessor a +1 luck bonus on saving throws, ability checks, and skill checks.
And the Combat chapter says (emphasis mine):
At the start of a battle, each combatant makes an initiative check. An initiative check is a Dexterity check. Each character applies his or her Dexterity modifier to the roll, as well as other modifiers from feats, spells, and other effects.