I want to have custom color picker in web part properties to let users select colors for headings. Any idea how to achieve this.
You can try to replicate a simple color picker with a dropdown control setting the corresponding background color foreach item. This way you can get the following:
Maybe this could be a good solution for your requirement.
How to create a custom web part editor: http://msdn.microsoft.com/en-us/library/office/hh228018(v=office.14).aspx
How to build a Color DropDownList control http://www.codeproject.com/Articles/19356/Multi-color-DropDownList-using-C