Some people think it means that an object, when it comes in contact with the skin, won't cause any adverse reaction.
But in fact, the truth is, it's relative. Because everyone's different, what is hypoallergenic to one person, isn't to another.
Some people think an item must not be hypoallergenic if it tarnishes when worn. This is not the case. All precious metals tarnish in certain circumstances, and each person's skin is different. Some people find all precious metals tarnish on them, others, only some. Others, none at all.
Sterling silver can go black on some and green on others. A silver bracelet may leave a green tinge on the wrist, for example. This is caused by the sulphur compounds on the skin and in the air. It doesn't mean there's nickel or some other toxic metal in the silver compound.
As far as children's jewellery is concerned, it is usually children's earrings that cause some sort of reaction. Some people are even sensitive to sterling silver and gold. In these cases, they are usually able to wear surgical steel earrings.
Occasionally baby bracelets in both silver and gold, can tarnish but this is easily fixed wtih a buff up using a Jeweller's polishing cloth, or your local jeweller should be able to clean it. Other items which can cause a problem is the tarnishing of sterling silver chains. Again, they can be cleaned.
When storing your outgrown baby jewellery, pop them in a small zip-lock plastic bag, and press out the air as you close it. This will keep out any sulphur compounds in the air and the items will stay like new for years.