for me , i prefer adobe illustrator
but it's kinda complicated & expensive for the average user or someone doing graphic design casually
there are several small & easy logo design tools such as " Sothinkâ€™s Logo Maker , LogoYes , Summitsoftâ€™s Logo Design Studio ... "
For Logo design I prefer to use CorelDraw. Basically everyone have their different choices. There are many software in market for same thing but Only one software is good in which you get grip and make logo easily..
Up*until i got Photoshop and Illustrator, which are the industry standard for professional design, i used Gimp. It is the best free graphic design software that i have used. I noticed vaneetagoswami mentioned CorelDraw, I have never worked with it so i can't comment on it. Give them both a try and see which you like best. If you can spend a little money, try Laughing Bird software. I think it costs around $50 and it has great out of the box templates and is very easy to use.
For me I prefer using Corel Draw because I feel I'm just like drawing on a paper and not having the layer-by-layer feel. You can make a vector art file as large as you want, very suitable in creating logos.
On the internet you can find tons of tools about logo, and they are working well. And the tools are: thefreelogomaker, logogenie, logogarden, logodesignengine, logaster etc. These are some free and some have free trial version. But in worldwide professionals use illustrator or corel draw for its unique feature, and these tools are not free.
I cant not say about any free tool or online, because you need logo for your website, so I think it must be unique and must be structure. I got info from my one of my acquit ant about an logo maker company and I haired they prepare logo in very small amount if possible for you, I can share their contact link with you if you wish to get it.. hope it will be helpful for you..