We have a small bird (wren?) that has made a nest in an old cat cozy (those cylindrical things a cat is supposed to sleep in to keep warm) in our garage. The nest has eggs. We know not to touch the nest, so we moved the entire cat cozy setup to a tall ladder on the front porch, with the bird inside. But she came right back to the garage looking for her nest…so we moved the cozy back. But I’m afraid the garage will get too hot and fry the eggs (Texas). How can we make it stick with her that her home is permanently relocated?
WAG - You can’t. Don’t relocate the bird nest but relocate the purpose of the garage. 
Until the kiddies are hatched and fly away, your garage is home to them. It was picked for a reason.
Keep the garage door open. Set up a fan? Move the car out?
Birds have high body temperatures and are pretty tolerant of high temperatures. The parent birds probably know what they are doing, and have selected a place that won’t get too hot. When it is hot, they may sit on the eggs, regulating their own body temperature by panting and thus preventing the eggs from overheating.
This said, there are some birds that do make mistakes in locating their nests. Although it may seem hard-hearted, it is probably best not to interfere with the Darwin awards. 
If the nest fails, they will probably re-build elsewhere.
As soon as I saw the title to the thread, I figured it was a wren. The eggs will be hatched very soon, so the temperature shouldn’t be an issue. Even in Texas, the garage is normally cooler than the outside. The problem that I can see is that if your garage door is closed, the mom can’t get in or out.
I had that happen several times to me when living in my own house-I usually felt honored when the Carolina Wrens decided to build a nest in one of my human contrivances. Usually the can squeeze underneath the side door or in general through the smallest imaginable cracks to get in and out. The garage may also keep predators out (no guarantee there tho).
This very thing has happened to a friend - Carolina wren’s nest in a spare bucket on a shelf in his garage. The best plan is to do nothing (except to take photos every so often). Birds are tough and persistent - if you leave them more or less to themselves, they’ll do fine.
As Colibri notes, temperature isn’t likely to be a problem. If it is - well, that’s natural selection in action (a minority of birds survive their first year - those that do tend to be well adapted to their environment).
Actually, this is rarely an issue. Few birds have much sense of smell, and (as noted) they tend to be quite intrepid in raising their young.
A couple of weeks ago, next to the field where my son was practicing soccer, I saw a couple of Killdeer hanging out, so I went to take a look. And there around them were three of the cutest little Killdeer chicks. You know how baby ducklings are so cute? These were smaller and cuter.
So I went near a chick to check it out - it had tiny little fuzzy stubs for wings. Did I mention how cute they were?
But when I was looking closely, I saw a behavior from the parents that I’ve never seen before. One of them landed on the ground about 10 yards away, and held its wing at an oddball angle, pretending to be injured, apparently in a strategy to lure me away from the chick. Each parent did this.
It was a good opportunity to talk to my son about how complex behavior can evolve, that even though the birds didn’t know why they were doing what they did, it was hardwired into their brain to do it anyway. It took some explaining, because he thought that the first of the Killdeer ancestors to do this must have been smart, so I attempted to explain to him that no, behavior similar to that was selected for and had gradually grown more and more similar to what an injured bird would look like.
Killdeer chicks and eggs are incredibly well camouflaged. Killdeer are well-known for their “broken-wing” displays to lure predators away from the nest.
Colibri and Xema answered the questions (both direct and implied). But as a sidelight–
Wrens often build multiple nests. She apparently requires him to give her a choice, and allow her to exercise the prerogative— she selects the one that will actually contain their eggs. So this nest may or may not ever be actively used. If you find that it never contains any eggs, be assured they chose another site nearby. It will almost certainly not be affected by your manipulations.