I guess it comes down to what “tinkering” means. I break firearms tinkering down into two broad categories. The first is replacing parts or adding accessories as with a Ruger 10/22. The second is taking a marginally or non-functioning firearm and making it function correctly. I’ve done both and enjoy both. The replacing parts or adding accessories tinkering takes more money and less skill. The making it function tinkering usually, but not always, takes more skill and less money.
One project I have considered is obtaining a low or non-functioning Remington 870 Express 20 gauge shotgun. There are several reasons for selecting a Remington 870 Express as the project gun. The first reason is that the 870 platform if correctly executed is a very good platform. However, Remington seems to have problems with consistent execution of 870 Expresses. Sometimes I think Remington should sell the Express as a “kit gun with some assembly and adjustment required”. The second reason is that parts, if needed, are reasonably priced and readily available. The third reason is that the mechanics and operation of the platform are simple and well known. The fourth and final reason is that 870 Expresses are readily available and if you can find someone with a “lemon” you may get it at a very good price. As a final note, if you totally pooch it you won’t feel bad about destroying a collector’s item.
Some of the possible work would include chamber polishing, barrel hood smoothing, action smoothing, and perhaps a Timney trigger kit. Depending on the end use there are other accessories that could be added once the gun is functional and reliable. This would trend more to the add accessories type of tinkering.
In summary, a low or non-functioning 870 Express bought cheap could provide a lot of tinkering opportunities and experience and when completed provide a good functional shotgun of greater value.