Fix the "Blogger Images Not Uploading" Problem
Clear Browser's Cache:
Start by clearing your browser's cache. Many browser related problems including pages that freeze, don't finish loading, or contain old content can be fixed by clearing your cache.
Disable any Plugins/Extensions on your browser. Plugins may cause incompatibilities with the post editor.
Enable popup on your browser or add an exception to blogger.com on your popup blocker. Sometimes popup blockers will prevent the image upload window from appearing.
Enable 3rd Party Cookies:
Make sure all 3rd Party Cookies are enabled in your browser.
Restart your browser:
Once you have tried the above three steps, you may also wan't to try and restart your browser.
Try a different browser:
Sometimes changing your browser may do the job for you. If you are using Chrome, try Firefox, IE or Opera.