Sad to say, Houston doesn't seem to offer much for early morning breakfast fine dining.
I'd skip House of Pies. It's not much different than IHOP. Actually IHOP's pancakes are better.
You might try 59 Diner. The locations are not that near the Galleria but I find them to be reliable and good quality.
17695 State Highway 249, Houston, TX 77064
Try the St. Regis Hotel dining room just off San Felipe about 2 blocks south of Loop 610. They have a wonderful pulled brisket and potato hash topped by two poached eggs served with an incredible Hollandaise. All of their breakfast offerings are very good, and they have one of my favorites, fresh ruby grapefruit juice! The space is lovely and even when crowded, tables are well spaced and it's quiet.
re: Ms Ghost
I stayed at the Ritz Carlton which is now the St. Regis, a wonderful property. I agree with FarleyFlavors on Le Peep, overpriced and bland. Let's see, Le Peep or St. Regis, Le Peep or St. Regis? I'll take the St. Regis, even if it costs twice as much, which it probaby doesn't.
Some possibilities (I haven't been to any for breakfast nor know hours but they do serve breakfast):
Le Peep - Houston chain, just a block inside 610
Patisserie Andre Tellier - about a 1/2 mile north of the Galleria or 2 miles inside the Loop
Kenny & Ziggy's - caddy-corner from the Galleria
There are websites for the other two.
"Great eggs benedict" or "quality diner" at that time in the morning might be tricky to find.
There's Mama's Cafe at 6019 Westheimer which opens at 0630. They do a passable Mexican Benedict which has chile con queso replacing the hollandaise.
House of Pies at 6142 Westheimer is open 24 hours. It's hardly going to win any awards either but the breakfast is more than acceptable. The O'Brien potatoes are tasty.
For a decent Benedict you might want to consider The Egg & I at 4938 Beechnut (it's a few miles south but should only take 10 minutes or so on the 610 at that time in the morning). Yes, it's a chain, but I reckon their hollandaise is actually quite good.
Edit to add: There's also Chacho's Tacos at 6006 Westheimer which does some good Tex-Mex breakfasts, also open 24 hours.