Here are three websites that let students create logos to use in design projects.
Online Logo Maker : http://www.onlinelogomaker.com/applet/logomaker/
LogoEase : http://www.logoease.com/LogoEase.aspx
LogoMaker : http://logomaker.com/
Each one lets you choose from ready made logos/shapes and then change and tweak the various components and colours and add your own text.
LogoEase and LogoMaker want you to register to get features such as saving – but you could bypass the need for this by screen grabbing the finished logos and pasting them into other drawing software. Online Logo Maker will let you download the logo as a png file without the need for registering.
All three use Flash so they are not suitable for use on iPads.
Nice but a little sneaky in regards to owning your logo. For example you can design and use something and someone else could then Trademark your design unless you do it first by paying them money.