Monk-ey is a pet acquired by using any prayer that grants Prayer XP on player attack during Combat.

This page is up to date (v1.2.2).
Pet ID: melvorF:Monkey
Source: Prayer
Drop Chance: See: Acquiring Pets
Effect: +5% Chance To Preserve Prayer Points
Part of 100% Completion: Yes

There is no best weapon to use in order to acquire Monk-ey - increasing/decreasing attack speed or using weapons with special attacks that perform consecutive attacks does not change the chance to unlock Monk-ey over time. Using multiple prayers at once does not increase the chance of acquiring Monk-ey, nor does the amount of prayer points consumed affect the chance in any way.