I was intrigued by your question since I asked myself the same before. So I started investigating what happens when you click on 'Add here'.
It triggers a webedit command: addrendering.
If you look at the Sitecore.ExperienceEditor.config in the commands section, you'll find the following:
<command name="webedit:addrendering" type="Sitecore.Shell.Applications.WebEdit.Commands.AddRendering, Sitecore.ExperienceEditor" />
I haven't looked at all the code in the AddRendering class, but there was 1 line in the Run method that got my attention:
SheerResponse.ShowModalDialog(dialogUrl, "720px", "470px", string.Empty, true);
720px is the exact width of the Add rendering popup modal...
So I copied the class to a test project, overwrote 720 to something larger, and it worked!
Tested with Sitecore.NET 8.1 (rev. 160519)
Note: I do agree with jammykam, updating CSS is probably the easiest way
Note2: I noticed that the Insert Page dialog works differently. Insert Page is a SPEAK dialog, while Add Rendering is SHEER UI.
To change the width of the Insert Page dialog go to the Core database and navigate to: /sitecore/client/Applications/ExperienceEditor/Dialogs/InsertPage/PageSettings/DialogSettings