How to edit a page that has no "Edit Page" menu option in SharePoint

Once you are at the page you want to add a web part to, add the following to the URL:

To change everyone's view of a particular page, append 

?PageView=Shared&ToolPaneView=2 to the URL.

To change your private view of a particular page, append ?PageView=Personal&ToolPaneView=2 to the URL (this will only work on pages that allow you to have a personal view - to see if you can edit a personal view of a page, go to the page and click on the top right context menu beside the "Welcome, <your name>" link to see if there's a menu option called "Personalise this page").

At the end of the URL, that is after the .aspx, put a '?' then the value below, to use more than one use an '&'

Querystring	Function
Contents=1	This is probably the most useful of querystrings. Using this on any "Web Part Page" will take you to what is called the "Web Part Page Maintenance" page (screenshot below) which allows you to delete web parts from both personal and shared views. This feature is especially useful if you have added a misbehaving web part. In this situation you can find yourself in a "chicken and egg" situation, whereby you can’t access the page to remove the web part because the web part you have added is causing an error on the page. Note: I have actually come across a situation where even this technique couldn’t remove a misbehaving web part, in this situation FrontPage becomes the only option.
Mode=Edit	This flips a "Web Part Page" into edit mode, its the same as clicking on "Edit Page"
Mode=View	This flips a "Web Part Page" into view mode, it’s the same as clicking on "View Page"
PageView=Personal	Displays your Personal view of a "Web Part Page", this is the same as selecting "Personal View" from the "Modify Shared/My Page" dropdown.
PageView=Shared	Displays the Shared view of a web part page, this is the same as selecting “Personal View” from the "Modify Shared/My Page" dropdown.
ToolPaneView=2	Shows the "Add Web Parts" toolpane, allowing you to drag and drop web parts onto a "Web Part Page".
ToolPaneView=3	Shows the "Search" toolpane, letting you search through the web part galleries before adding a web part to the page.

Finally (there is always a finally isn't there?) a couple of things that should be considered:
1. These querystrings *will* respect the underlying SharePoint permissions, so for example, someone without the ability to add web parts will not suddenly be able to by using these querystrings.
2. These will let you add web part to pages that may not have been designed to have web parts added to them, for example the "Sites" page. Be very careful as there are certainly some issues to be discovered, don't try and get too fancy!
3. You use these at your own risk.

If you don't want to start filtering using a Form Web Part then using FilterField / FilterValue pairs is a nice easy route: 

?FilterField1={column name}&FilterValue1={required value}