Add spelling suggestions

+  /**
+   * The spelling suggestion
+   * @var array
+   */
+  private $suggestion = NULL;

Since that property is initialized to NULL, its type should also include that.

+   * Get Spelling Suggestion.
+   *
+   * @return array The extra parameters to CSE.

The return value description goes to a new line.

+    // save spelling suggetion
+    $this->suggestion = (string)$results->Spelling->Suggestion;
+

Comments are sentences: They start with a capitalized word and end with a period.
suggetion is misspelled.

+    if(!empty($spelling_suggestion) && is_string($spelling_suggestion) ) {
+      $html .= '<span>' . t("Did you mean ") . l($spelling_suggestion, 'search/gss/'.$spelling_suggestion)  . '</span><ul>';
+    }

There must be a space after if.
There is an extraneous space between the closing parentheses.
Strings passed to t() needs to use placeholders.
I am not sure it is necessary to use l() in this case.

I would also leave fixing any coding standards issue to a different issue. This is about implementing a feature and it should be limited to that. At least, the patch would still be valid even when the coding standards are fixed.

I am not sure the maintainers will add this feature, though. The project page says:

We discourage use this module, since Google decided to discontinue Google Site Search as a paid service.

If Google Site Search has been discontinued, new features cannot be used.