If you get too close to the logo image,
click on the clone brush and replace any areas.

Now promote the background to a layer so you can remove the white background color.
Layers/Promote background layer

Click on the magic wand. Now click on the background color.
O ne section at a time if needed. Click on the cut icon (ctrl-x) to remove the background.

Like this. Continue till the white is removed.

 

The lines need to be defined a bit more.
CLick on the eyedropper to get the logo color.
CLick on the pen icon. Below are the setttings I used.
Just add a line on top of the one already there.

Add a layer and drag it under the image layer.

You now have a clean logo where you can add any name you want.
Save as a psp file so you can change it as needed.

To make your base logos easier, check out
The Logo Creator
A nifty program.