* Heirloom: This is the correct and most likely intended word. An heirloom is a valuable object that has belonged to a family for several generations. It's often something with sentimental value as well as monetary worth, like jewelry, furniture, or a painting.
* Hair loom: This is less common and rather nonsensical. A loom is a device used for weaving fabric, so "hair loom" would imply a loom specifically designed for weaving hair. While theoretically possible (e.g., in a fantastical context), it's highly unlikely to be the intended meaning.
Therefore, if you're trying to understand what someone means by "hairloom," assume they mean "heirloom."