On page 149 of the PHB, you'll see a list of weapons (no doubt you've seen it). The morningstar is listed under Martial Melee Weapons and has no properties. Anyone can use a morningstar, but they require proficiency in martial weapons to add their proficiency bonus to attack rolls with one.
Weapons can be used for two-weapon fighting if they are one handed and have the light property. A weapon is one-handed by default, unless it has the "two-handed" property. Versatile allows you to use the weapon with either one or two hands. The morning star has neither of these properties, so it is one-handed, but it is not light, so it can't be used for two-weapon fighting unless you have the Dual Wielder feat. Two-weapon fighting is one way you can "use it to attack as a bonus action or something", but again, you require the Dual Wielder feat to use it. However, there is nothing limiting you from holding a non-light weapon in both hands and choosing which one to attack with for your attack action.
Others have mentioned to keep it for skeletons because they are vulnerable to bludgeoning damage, but a morningstar does piercing damage. You can keep it if you want to use it with a shield or if you want to keep your other hand free for some reason or if you want a source of piercing damage compared to your greataxe's slashing damage.