A Fossil hamster
Creating fossils takes time and patience (obviously). The best results are achieved over one hundred thousand to a million years or so, though if you are in a hurry there are options for creating a decent fossil over a mere thousand years or so.
Fossils come in many forms.
Now the quality of the fossil is controlled by two things, the surrounding rock, and the minerals that do the replacing.
Imagine that you have a perfect mould. Silica and calcium based minerals crystallise with very small, fine crystals, and therefore will fill the mould most perfectly and leave you with the best fossil. Pyrite however (fools gold) has very coarse crystals, so however good the mould is, the poor fossil will be no better than an indistinguishable lump. Normally the surrounding rock influences the minerals that are seeping through it, so limestone generally has silica based fossils and metal rich muds produce pyrite fossils.
And the mould is of course made up of the stuff that was hanging around while the animal died. So ideally to make a mould, you need to have the animal in as good a condition as possible, and you really need the stuff it dies in to be of a very soft fine texture that will bury it quickly and form a good mould around it. This is why mud is such a good place for an animal to die who wishes to be a fossil, and why shingle and gravel is probably the worst. Mud wraps the animal up away from the micro-organisms that munch it up very quickly, while gravel is all full of holes which can be like microclimates to the micro-organisms, and give them a lovely little environment in which to happily munch your aspiring fossil away before it so much as reaches a year, let alone a millennia.
(Incidentally, sand is actually quite good as a preserving material, in that it compacts down very well, and is associated with dry atmospheres which aren't conducive to rotting organisms. However atmospheric conditions and predators alone generally reduce the animal to bone before it is buried, and it is this reason why the dinosaurs are well preserved, but a trifle anorexic).
So. What you should do is club your hamster to death very carefully, and lay it out in a still mud at the bottom of an undisturbed sea, and cordon this area off for a few hundred thousand years. This will guarantee a good fossil out of your beloved pet (assuming you can remember where you buried it).
If time is ticking however, peat bogs and high mountains are always an option.
Peat bogs, being vaguely acidic and therefore devoid of those tissue devouring bugs, is a wonderful place to bury your hamster if you want a fossil in around 1-2 thousand years. Better still, NO tissue replacement occurs, what you bury is what you dig up and Hammy will still have its own teeth, hair and limbs (which is more than can be said about Grandpa), though he may be a bit brown and flattened and look a bit like a very old leather shoe. Don't worry, whatever state he is in, the archaeologists will go apoplectic with joy.
The other quick fix method is to freeze Hammy in ice and bury him in a permafrost somewhere. This could either be a glacier, or left around in a tundra environment, or even on a high desert mountain slope (such as the Andies) if you wish. Again excellent results are obtained over 1-2 thousand years, though Hammy will rot quickly when defrosted as - unlike the peat bog hamster, it has not been soaked in preservative acids for years so the bugs will soon start munching.
Please note, however! Whatever method you choose, don't neglect to include some interesting artefacts with your Hammy just to give added interest when he is finally dug up. A belt, a piece of string, an arrow head, some pottery, and of course the obligatory thingie that no-one recognises as anything useful and so calls it ceremonial...