I think on eBay and other marketplaces, it's OK to sell a range of items. However, if you want to sell on your own website (which I believe is what most sellers should work there way up to doing if they are serious about building a long term business), you should focus on a particular product market.
There are a couple of reasons for this but the main one is search engine optimization. If you sell everything, how will people find you? No one is going to run a Google search for "Bob's Emporium of Everything You Want!" or even something more generic like "one stop shop".
However, if you sell a specific range of items such as children's sportswear, you'll get more traffic from the search engines.
In general, I'm a huge believer in branding and I think a good brand sells products and allows the brand owner to charge more for those products too. It's hugely difficult to create a brand that represents a range of products (yes, WalMart do it, but let's not pretend we have the capacity to compete with WalMart!).
However, again, if you have a specific product range, you can build a brand and a following for those products.
If you do want to sell a range of products because you know they all sell well, you could always create different websites or eBay stores for each of these markets.
I've got a new niches under my belt but I would never merge them all together because it would look totally unprofessional if I did.
I hope this helps :)