For a job like that a standard DIY shop style pop riveter which comes with suitable rivets will be just fine - nothing special required. I think that mine is Draper or similar.
If you have not used one before then just remember to keep the nose of the riveter pressed up firmly to the material to which you are riveting, otherwise the rivet may end up loose - you may need some backing when pushing against something thin like a garage door to avoid denting it. You will be left with a hole in the centre of the rivet, but I assume that that does not matter to you.
I have recently purchased an Eclipse riveter having been very disappointed with my older Faithfull version, the Eclipse one is considerably better not jamming once which was the problem with the Faithfull one. Well worth paying a little more.
I bought a stanley one from B&Q a good few years ago for not a lot of money.
Had quite a bit of use, and pulled a fair few rivets bigger than perhaps it should.
I normally find I bend the handles on things like that first, but its been okay so far.
As above, keep the tool pushed in tight, and use the right length rivets.
Aluminium rivets should do it fine if its unpainted, and will be easy to pull up by hand.
Id think 4mm x 7-8mm long would be about right.
If its painted or requires a bit of strength, id go with steel ones.
Regarding patching material, if its painted, get a bit of 'zintec' sheet. This is a reasonably readily available type of sheet, with an elctro zinc coating, a light rub down, and it can be painted.
If its just a galvanised door, get yourself a bit of galvanised sheet.
Most steel fabrication/sheet metal shops would have a few offcuts kicking around probably. If its a large strip, eg full width, it might be easier to ring first.
I've had a variety down the years. I've a Stanley handled one that used to be good but now slips - pretty much worn out, I think.
I bought a Draper concertina-type some years ago, and it's been brilliant - well worth the extra money. It's very adjustable for pull, and does the whole rivet in one bite. I still make up a lot of electronics boxes, and rivets are great for fixing down transformers and fixing XLR (Cannon) sockets into chassis. This one does them nicely in one, controllable pass. For small work, I put the handle on the floor and push the work down onto it gently.
Strip and clean the collet jaws often with a wire brush, and use a fair bit of 3-in-1 oil in the collet. It won't stop it gripping, but it will help it release the pin - the handled ones need to do the grip-pull-release several times with standard length rivets, and if they pull too much each time and grip too hard they tend to jam, which is a right PITA.
Don't be wary of using washers on the back. It's often a bit of a fiddle to get them in place, but they make a huge difference to the strength of the finished joint, especially with thin steel and aluminium plate. The closer you can get them to the rivet diameter the better, placed so the sharp-edged side of the washer faces towards the plates you're joining (and the rivet gun).
Finally, if the finished look is important, pull on the riveter, away from the work, gently, as you do the last squeeze. That way, when the rivet parts, the pin won't bounce onto the surface, ruining any new paintwork. DAMHIK!
Pop rivets are so much easier than fiddling with small nuts and bolts. Have fun!
PS: from good fastenings suppliers you can get steel or aluminium rivets. Steel do take quite a bit more effort for any given size, and I wouldn't attempt them with an ordinary hand riveter. If it's galvanized sheet, aluminium will probably be just fine, as long as it can't abrade when the door opens, etc. If you're fitting a strengthening plate around the lock, I'd fit a backing plate and bolt through both, as pop rivets won't withstand a crowbar (again don't ask!).
I've recently fastened some Screwfix draught proofing strip to the underside of my garage up and over door - using my friend's Halfords pop riveter.
Worked well. It was to keep mice out rather than the draughts and so far no more mice!