I'm not even sure I can make sense of the notion of obstinate rigidity. My main purpose here is not to discredit it, however, but to do something more modest: I want to suggest that a line of interpretation of Kripke, which has him sometimes working with a notion of obstinate rigidity instead of his normal official notion of rigidity, is mistaken.
From the Stanford Encyclopedia of Philosophy article 'Rigid Designators':
In other places, Kripke seems to have in mind another account of rigidity: one according to which a rigid designator designates its object in every possible world, whether or not the designatum exists in that world. Hence, he says, “If you say, ‘suppose Hitler had never been born’ then ‘Hitler’ refers here, still rigidly, to something that would not exist in the counterfactual situation described” (Kripke 1980, p. 78).The interpretative suggestion before the quote seems wrong to me. I find it more natural to think that Kripke is saying that 'Hitler' actually refers, rigidly, to something that would not exist in another counterfactual situation - not that it refers to that thing in that situation.
Here we have to be careful to distinguish:
What does expression X refer to in counterfactual situation Y? (Which in turn must be carefully disambiguated, in a familiar Kripkean way, so that it does not get interpreted as asking about how language would be used in counterfactual situation Y.)
What does expression X actually refer to when it appears in this description of counterfactual situation Y?
You can answer 'Nothing' to the first and 'Hitler' to the second.
(Things may get even subtler when you consider definite descriptions. Suppose John is the tallest man in the world, but Paul might have been. Then if we say 'Paul might have been the tallest man in the world', should we still say that the actual referent of 'the tallest man in the world' in that sentence is John? To this, one wants to object that John isn't really involved. (But does that matter, when the question was about what the actual referent is?) Perhaps someone could find principled grounds for denying 'The actual referent of "the tallest man in the world" in that sentence is John' while accepting the 'Nothing'/'Hitler' pair of answers above. Perhaps Donnellan's distinction between referential and attributive uses of definite descriptions, or something like that distinction, should come into play here. Still, this complication - and it really is enough to do your head in - shouldn't impinge on the plausibility of what was said above.)