How to export to HTML and not lose Grid Frame in the generated HTML?

Following my comment on your question, if you add

table
{
border-collapse:collapse;
}
table, td, th
{
border:1px solid black;
}

at the beginning of the css file, you will get:

P.Fonseca's answer will work fine as long as you don't mind editing the CSS file after export.

There might be another way using the ConversionRules options for stylesheets.

Something like:

mystyle = 
Notebook[{Cell[StyleData["GridBox"], 
 ConversionRules -> {"HTML" -> {{"<table border='1'>", 
      "</table>"}, {"<td>", "</td>"}}}]}];

I have not yet got it working as it should and you will probably still need the border-collapse:collapse; CSS setting on the table element. But this might point you in the direction of an automated solution.