
ANSWER: When a person, such as yourself, inherits someone else’s collection, they need to decide whether to merely curate the collection, that is take care of it and preserve it, or to make it there own. It sounds like you’d like to make your father’s matchbox collection your own. The first thing you need to do is learn about the history of these unique containers, then you need to find out which types are the most collectible, not necessarily the most valuable.

Before 1844, when Gustraf Eric Pasch invented the safety match, finding a source to light a fire in an emergency was a challenge. He devised a system of impregnating little sticks of pinewood with sulpher and storing them ready for use. At the slightest touch of fire they burst into flame. Formerly called a "light-bringing slave", it later became known as a “fire inch-stick.”
But it was Edvard Lundström who developed Pasch's idea of a safety match and applied for its patent with a phosphor-free tip. Johan's younger brother, Carl Frans Lundström was an entrepreneur and industrialist who helped him set up a safety match factory in Jönköping, Sweden, between 1844 and 1845. They began making matches in 1853 and won a silver medal for their invention at the World Expo in Paris 1855.
Although expensive to produce, their matches became known throughout the world as Swedish Matches.

In 1892, Alexander Lagerman invented a machine that revolutionized safety match manufacturing. The

In that same year, a brewing company ordered more than 50,000 matchbooks to advertise a new product, thus the practice of matchbook advertising was born. Once they became common, advertisers were eager to use these popular items to get their messages to the public.
Advertisers display a wide variety of both consumer and industrial goods and services using matchbox ads. However, restaurants clearly dominate all other categories of trade. Next in line are probably hotels and motels, yacht and country clubs and other types of membership organizations; industrial firms, retailers and financial institutions. Represented to a lesser extent are food products, liquor, tobacco, tourist attractions, transportation, mostly airlines, real estate, insurance, automobile dealers, sports, public utilities, and governmental agencies.

Some matchboxes can be personalized with a loyal customer's name imprinted to reward patronage in restaurants and other businesses. These often have a few blank lines printed on the back so that the user can note names, addresses, phone numbers and notes.

While manufacturers used plain cardboard for the majority of matchboxes, there are some made of glossy coated cardboard, foil, plastic, and wood.
Besides the United States, collectible matchboxes can also be found in other countries, such as England, Canada, Japan, Australia, Korea, Italy, Austria, Hungary, Sweden, Spain, Ireland, Wales, and France.
Matchboxes are an affordable collectible with many examples selling for mere pennies. There’s also a great deal of variety with over 250 different matchbox categories such as military or hotels. While the U.S. matchbox collectors is facing a diminishing supply because people are quitting smoking or using lighters, the foreign hobby is still going strong. Diligent U.S. collectors can also still find giveaway matchboxes, however.
To read more articles on antiques, please visit my Web site. And to stay up to the minute on antiques and collectibles, please join the other 17,000 readers by following my free online magazine, #TheAntiquesAlmanac.