Gift-giving is a well-honed art in the Shire; a socially responsible Hobbit will spend prodigious hours in the days before his or her birthday deciding upon the presents he or she will give to friends, family, and enemies.
Of course, Hobbits are just like you and me in many ways. While preparing for his or her birthday a busy hobbit may become pressed for time, and will not devote the same amount of thought to a third cousin as to, say, a favourite auntie, or the person who eloped with his or her fiancee. Not only that, birthdays are a prime opportunity to rid the smials of useless clutter. A prime class of useless clutter being the gifts from the birthdays of distant relatives, businessmen soliciting custom, and the like.
And thus it happens that certain objects are passed from hobbit-family to hobbit-family to hobbit-family dozens of times. These are mathoms.
Some mathoms acquire quite a reputation as they pass down the generations. The largest town in the Shire, Michel Delving, contains a "Mathom-House" (museum) with some of the most famous. Of course, The Hobbit and The Lord of the Rings specify but one of the exhibits from that estimable establishment: The suit of mithril-mail acquired by Bilbo Baggins while he was away lollygagging in the East.
All right, all right. Recall that J.R.R. Tolkien was a linguist before anything else, and his stories contain all sorts of linguistic treats. Tolkien evolved the Anglo-Saxon word maðm (meaning "precious gift" or "heirloom") into this archaic-looking but nonetheless modern-English-looking word. If you're expecting a Webster definition to appear below with an [Obs.] in it, you're in for a disappointment.