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It is possible to open the "Run..." dialog by using the keyboard shortcut (Windows-Key+R).
It is also possible to create a normal shortcut with an icon that points to the "Run..." dialog.

1. Create a text file named Run.js, which contains the following line:

   (new ActiveXObject("Shell.Application")).FileRun();

2. Right-click Run.js and select "Create shortcut"
3. Right-click the created shortcut and select "Properties"
4. Add "WScript.exe " (Without quotes) to the beginning of "Target:"
5. If wanting to change the icon of the shortcut press "Change Icon" else press "Ok"

Note the same technique can be used to launch the Search dialog:

(new ActiveXObject("Shell.Application")).FindFiles();

Note if using Windows XP, then one can also create the icon by dragging the "Run..." icon from the Start Menu to the Desktop (Doesn't work in Classic Mode).

Credits The Old New Thing