Although you can't directly edit the standard search boxes, you can set the height through CSS. Assign the style "ms-sbplain" the height (and any other cosmetics) you want the search box to have. The simple way to do this for a single page is via adding a Content Editor web part with a style segment:
Code Snippet
<style>
.ms-sbplain{
height=50;
background=#ffff00; }
</style>
This sets the height to 50 pixels, and the background color to yellow, overriding the default style from core.css or any theme you have applied. (Note: you can also incorporate such a segment into the custom theme or other branding .CSS file your site uses, so it automatically applies to every search box.)