Must the trademark have a connection with the product to which it refers?
The purpose of the trademark is indeed to enable whoever buys a product to immediately recognize the producer or trader, and, on the other hand, to ensure the producer to be easily recognized by his customers. For this reason, even if a trademark often gets registered simultaneously with the introduction on the market of a new product, its function is more important if the product is ordinary and of little innovation.
As a better explanation, if an innovative product, previously non-existent, is sold, like the famous Rubik cube, the seller will take over quite a big slice of the market even without registering a name of effect for it as the item itself has quite strong features making it easily recognizable. If instead I produce traditional objects, the trademark (and the design too) will be of primary importance to attract customers and also to promote myself on the market, therefore it will certainly be convenient registering a trademark.
Regarding the name to be chosen, it is useful to bear in mind that it must not simply be a different definition of the object, but it should be an imaginative name, a curious name, which will only indirectly refer to the item and because of this distinctiveness, will involve and attract attention.