People collect baseball cards, stamps, egg cups, coins, toy soldiers, coupons, theatre tickets and much more besides. Unusual things I have seen collected are butterfly wings (how cruel), insects (yuck), songs (vocal), birds (in cages), furniture (too big), glass (dangerous), guns (even more so) and buttons (why?) but I am sure that you can find other collections, strange and not so strange.
Small items are generally easier to gather together and the storage of them is much less troublesome and less space consuming. If you are restricted for space then you might consider things like miniature cars, books, stamps, coins, toy soldiers or thimbles and so forth. If you have a lot of room then there is no restriction (other than money) and you might collect life size cars instead of miniatures.
A bit of research will be needed before you begin to collect anything but start with something readily available like postage stamps. A great variety is available in your everyday mail and two countries whose stamps offer great variety are US postage stamps and Canada Post but no matter where you live you will have a ready made supply of your local stamps. It is not an expensive hobby to start with but you will find many collectors are people that just gather what others throw away.
If stamps are too flimsy for you then why not try coin collecting. Some currency collectors will collect notes only but this is a bit flimsy also unless they are protected and kept in mint condition. Glass is often collected in bottle form and some bottles are quite decorative although I use fancy bottles as candle holders, indeed there are many ‘candle’ collectors too. Whatever hobby you choose will be yours alone and no matter how ‘odd’ it may seem to others I guarantee that you will find many other collectors with whom to share your interest.

