I should start by saying that I am not a lawyer. I have been both the complainant and defendant in patent cases, and have had the role of observing copyright compliance for a performing arts organization. With that in mind, the following is my own opinion and information.
Ultimately, there is no simple answer to your question. It would depend on the case law that applies to 3D printed objects, which is not very clear. You are venturing into Copyright law, which is very different than Patent law.
For example, if someone had a patent on "A Raspberry PI case with a <describe a novel, special, functional feature>", and you made a case with that feature, you would be in infringement. It wouldn't matter if the case looked like the original, or was completely different. If it included that patented invention, you would be more likely to lose if challenged in court. The one thing about patent court prosecutions is that it is really rare that anyone actually "wins". The cost to put forward a case is very high, and usually someone runs out of money (sometimes even the "good guy") before a decision is reached.
Copyright is much harder and softer at the same time. Copyright can relate to the design feel of an object -- such as rounded vs. square, or using a trash can vs recycle bin icon. Prosecuting a copyright violation of this kind would require that the aggrieved rights holder demonstrate that the design was copied, or at least derived in an unpermitted way, from the original work. This is often more a matter of opinion than law, which is why the lawyers matter, as well as the judge and potential jury.
Technically, you are not free to do whatever you wish in the privacy of your own home. You are much less like to be detected and then prosecuted, and the penalties would be lower, but you are just as much in violation if you make one for your own use or sell them by the millions.
The best way to handle it, if you are prepared for possible adverse reaction from the original designer, is to ask them for permission. You can say ask for clearance to use, such as by saying:
I loved your object <thing> on Thingiverse. I plan to <put in your plans here -- make a few for me and my friends -- sell further varieties online -- whatever>. Although I didn't copy your design, I would like to acknowledge that I saw your design before doing my own. How shall I do that?
Your next steps will depend on how they respond. Most likely, they will either give you free leave to do as you wish, or they will ask for something.
This happens a great deal in music, and expensive and acromonious problems have developed over who actually composed a particular guitar riff. You can pick up the "Stairway to Heaven" case here. If you are doing this commercially, it is better to negotiate ahead of time. If you are putting your object back on Thingiverse for sharing, I'd just identify it as derivative an get on with life.