These mutations are an important way that a genome can get larger, and hence contain more information.
Let us imagine a sexually reproducing species. Some gene G contains the information about building some protein P. Some individual accidentally acquires a second copy of G. If that isn't harmful, the mutation could spread through the species, and by neutral drift it could become the norm, or at any rate common.
At some later date, another individual has a mutation to one of his two G's. The altered gene now causes protein P2 instead of protein P. Suppose that P2 is different from P in a useful way. The individual still manufactures P with his (or her) undamaged copy of G, but now P2 is manufactured too. The individual now contains new genetic information. Since P2 gives an advantage, the new information has a very good chance of spreading through the species. The species eventually comes to rely on P2's special properties.
Essentially I am saying that a gene can get copied. And then, the two copies are free to diverge - to mutate and become different - without the creature promptly dying from loss of the old function. Occasionally, the copy that diverges becomes beneficial.
The new protein P2 doesn't have to be wildly different to be useful. For example, hemoglobin is the molecule which carries oxygen in your blood, but a human fetus uses gamma-globin instead. The two molecules are very similar, but the adult and fetal environments make slightly different demands.
Could this scenario happen? Yes, because duplication mutations are observed today:
"Actually, the process of gene duplication can occur in a number of ways, and the most common mechanisms are well understood. Sexual organisms, for example, have two sets of chromosomes (one from each parent) which line up during the cell division process called meiosis. As it happens, the very long DNA threads are constantly breaking and being rejoined. The rejoining process is not 100 percent accurate, however, and often one of the chromosomes comes away with a little more of the DNA than its pairee, which will have correspondingly less. The lucky gametes that come away with the more are said to have had a "gene duplication," although the amount of DNA may amount to only a part of a gene or maybe a whole string of genes."
Russell Doolittle, Boston Review, Summer 1996
Do duplications harm the individual? Well, they can. Some cancers seem to be caused this way, and the Charcot-Marie-Tooth neurological disease seems to be caused this way. Thalassemias are caused by having the wrong number of hemoglobin genes. (Some humans have more than normal, and others fewer.) Duchenne dystrophies and cystic fibrosis come from deletions, the opposite side of the coin from duplication.
But some duplications don't seem to matter much, and some are known to positively benefit the possessor. So the scenario isn't asking for anything unknown or particularly unlikely.
Is there evidence that this scenario has happened? Yes, it seems to have happened often. Myoglobin, which stores oxygen in muscles, strongly resembles hemoglobin, which carries oxygen in blood. Both are necessary to humans, but invertebrates such as the worm C. elegans only have one kind of globin. So, it is a reasonable hypothesis that our genes for myoglobin and hemoglobin are "descended" from one single ancestral gene, which got duplicated. And, in fact, there are "unnecessary" structural similarities between human myoglobin and human hemoglobin. If the similarities aren't due to common ancestry, then we have found a really really big coincidence.
Lactalbumin is a component of milk. It and part of the lactose synthetase enzyme are encoded by genes that differ only slightly from the gene for lysozyme, which retards infection by breaking down bacterial cell walls. So, a duplication and a little tinkering can produce useful new objects from odds and ends.