Maybe not a fallacy, but...
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Original Question
I was wondering if anyone knows the name to this kind of strategy that someone does when someone demonstrates that there is an inconsistency in one of the concepts used among the premises of his argument, for example:
Person 1 asserts x
Person 2 proves that x is inconsistent
Person 1 now asserts x' (where x' is just a variant of x to evade or escape Person 2's argument).
Person 2 proves that x' is still inconsistent or presents new inconsistencies.
Person 1 now asserts x'' (where x'' is just a variant of x' to evade or escape Person 2's argumentation)
Ad infinitum
The best-known example of this case is that of the Dragon of the Garage used by Carl Sagan to show that theists often resort to this strategy when applied to God. This type of strategy makes it impossible for Person 2 to ever show that x is inconsistent because Person 1 will always be able to redefine the term over and over again.
Comments on Question
I think you have hit upon something important here. It is one of the biggest problems I see in attempting to use logic in debates. What several commentators have called this is "moving the goalposts," but that suggests a formal debate or other arena where points scoring is important. In most of these cases, the debate is an attempt to convince or persuade someone a strongly held belief is (in part or in whole) illogical. In such cases, motivated reasoning causes the person upon whom you are attempting to use logic to be 'slippery' with his/her definitions -- as slippery as is needed to avoid admitting the point you are making is correct.
Sometimes this 'slipperiness' occurs in small ways (since all words have ambiguous connotative meanings) and sometimes it occurs in much larger ways (such as completely redefining words). I imagine this is how we got to the situation where if you look up the word "literally" in the dictionary you will find it can mean the exact opposite of its original and obvious definition (i.e. it can mean "figuratively").
Using your example, imagine trying to debate someone who asserts X is literally true, you show that it is not true (or inconsistent with the facts) and then that person responds they meant it was figuratively true since "literally" can also mean "figuratively"!
I'm not familiar with Sagan's example, but thanks for sharing it. I will look it up. As a Theist, I have also experienced this problem in debating with Atheists. Indeed, I had a debate with an Atheist on this platform that consisted in part of debating the proper meaning of almost every key word that was used. It was quite irritating, so I can see from where you are coming.
To answer you properly, we would need to shed some light on the relation between x and x'. In any case, someone who speaks in this way would quickly bring the opponent to the edge of his nerves.
Answers
3This appears to be an example of moving the goalposts . Perhaps the difference might be instead of more "points," they are presenting "variants." Same form of fallacious reasoning would apply.
'"The best-known example of this case is that of the Dragon of the Garage used by Carl Sagan to show that theists often resort to this strategy when applied to God.".......this just shows that Carl was logically inept. Dragon of the Garage isn't analogous to theists "doing" anything with God. For one, the Dragon idea has a predetermined environment as to where one looks for evidence. The second and more important idiocy of The Dragon's non'analogy presupposes that theist's definition of God isn't predetermined. The example or straw man misrepresentation Sagan uses is idiotic and anyone who "believes" theists do that makes a borderline hasty generalization about theists or they're just ignorant.
Among others, moving the goalposts strikes me first. This is an argument in which a person dismisses counter-evidence presented in response to a specific claim and demands some other (often greater) evidence.
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Prove to Person 1 that his assertion is inconsistent, and you satisfy him for that round of the discussion. Teach him how to assess his claims for consistency, and you satisfy him for the rest of his life and you can stop arguing with him ad infinitum.