Why were the Hecatomnid rulers of Caria permitted to marry their siblings?

Short Answer

There is no clear primary source evidence for either why the Hecatomnid dynasty (c. 395 - 334 BC) siblings married each other or why it was 'permitted' (by which I take it to mean why the citizens didn't drum the siblings out of town), but the former question is discussed in some detail by E. D. Carney in Women and Dunasteia in Caria. While noting the lack of scholarly attention that has been paid to Hecatomnid sibling marriage, she argues that

The Hecatomnids probably turned to brother-sister marriage in order to elevate the status and establish an identity for their new dynasty.

As to why this was 'permitted', none of the sources seem to deal directly with this. The answer may well simply lie in the political and economic benefits of stability, and the desire of the populace to keep a satrap who was local rather than Persian. The father of the children in question, Hecatomnus (ca. 395–377 BC), who succeeded the Persian Tissaphernes as satrap, was the first non-Persian satrap under the Achaemenids.

Carians, although heavily influenced by Greek culture, were nonetheless regarded as 'barbarians' by the Greeks. Being firmly under Persian control by the time the Hecatomnid dynasty came to power, the elite were naturally oriented more towards the east in their political outlook, and in the east such marriages were already in evidence (as they were in Egypt, with which Caria had long had links). However, to what extent (if at all) these factors influenced the opinions of the local population is impossible to determine with any degree of certainty.

Further, as Mark C. Wallace points out in his comment above, even if "nobles, soldiers, merchants and ordinary citizens" did not approve, we are talking about an externally (Persian) appointed autocrat, not a democratically elected leader; the effort required to remove the former is much greater than that needed for the latter.

The Persian kings were primarily concerned with the satraps sending their taxes on time, maintaining peace and responding to occasional requests to march against troublesome neighbours. If those obligations were met (by and large at least), the Great King was probably not concerned with the domestic arrangements of his minions.

Details

First, an outline of the Hecatomnids (father and children) seems in order:

• Hecatomnus, satrap of Caria circa. 395 to 377 BC and father of the following:
• Mausolus, eldest son, satrap from 377 to 353 BC. Husband of his sister
• Artemisia II, satrap in her own right from 353 to 351 BC but with likely junior co-ruler status during the reign of her husband. She was succeeded by her brother
• Idrieus, satrap from 351 to 344 BC and husband of his younger sister and successor
• Ada, satrap in her own right from 344 to 340 BC but with likely junior co-ruler status during the reign of her husband. Restored in 334 BC by Alexander the Great after she been usurped by her younger brother,
• Pixodarus, satrap from 340 until his death in 335 BC.

One of the Seven Ancient Wonders of the World, the Mausoleum at Halicarnassus, was built for Mausolus and Artemesia II.

Given the lack of evidence for the Hecatomnid marriages (other than who married who), Carney turns to discussions on sibling marriages in other dynasties and finds that:

The most compelling argument about sibling marriage made in terms of other dynasties has been that it is a strategy often pursued because it makes a royal family distinct from the population as a whole, and, at the same time, it imitates the marriages of gods.

Applying this to the Hecatomnids, Carney elaborates:

Why did the Hecatomnids, apparently without any compelling precedent, suddenly turn to this comparatively unusual practice? They were a new dynasty (in terms of rule of all of Caria) and these marriages made them distinctive immediately, much as the sibling marriages of new pharaonic dynasties had done. If both sibling marriages were arranged around the same time, by Hecatomnus, then the effect would have been even more intense. Just as the Mausoleum implies that Mausolus and his dynasty were heroic, perhaps semi-divine..., so might the unusual marriage habits of the Hecatomnids. They may also have been trying to buy authority by imitating occasional Achaemenid practice, although...this is not likely to have been a major factor.

Carney is almost certainly right to downplay the 'Achaemenid practice' as an influence because, during this time (but unlike the later Sasanian period),

the evidence for next-of-kin marriage, which was perhaps not in its origins a Zoroastrian custom, is supplied solely by the Graeco-Roman world. For these centuries there is no Iranian evidence.

Source: Joan M. Bigwood, ''Incestuous′ Marriage in Achaemenid Iran: Myths and Realities'. In 'Klio' 91(2):311-341, December 2009.

As to what the Hecatomnid's subjects thought of these marital arrangements, we have no surviving evidence. We can, though, reasonably surmise that this 'family enterprise' was planned by their father, Hecatomnus, and that his authority at least helped to discourage - as far as we know - any meaningful opposition. He had been appointed satrap of Caria apparently to help combat the troublesome military activities of the the Spartan king Agesilaos II and governed for around 18 years, apparently to the satisfaction of both his master (the Persian king) and the Carians.

Hecatomnus was from Mylasa and thus a local ruler, not an outsider (i.e. Persian). Thus, Carians were in a unique position at the time of being the only satrapy not governed by a Persian but by 'one of their own'. Disputing Hecatomnus' succession plans (if, indeed, they were his) would not have been in the interests of either the Great King or the local population, economically or politically.

On the Greek influence and their "revulsion towards incest", the Carians had other influences. In short, it's complicated, and the Carians were most likely a mix of Greek settlers and native Anatolians:

The extent to which this spatial perception of Caria can be equated to a sense of a Carian identity is complicated. Indeed, a coherent and unified conception of the ‘Carians’ should not be envisaged...

Source: N. C. Unwell, 'Caria and Crete in Antiquity: Cultural Interaction between Anatolia and the Aegean' (Cambridge University Press, 2017)

They were not well-regarded by the Greeks, with authors from Homer to Strabo (and many in between) writing critically about them. In fact, they were the target of several proverbs and heavy negative stereotyping:

Such negative stereotypes of the Carians are based on their perceived inferiority as non-Hellenes....Another saying, preserved by Diogenianus (active in the second century ad ), reinforced the notion of Greek scorn: ‘the Lydians are bad, the Egyptians come second, the Carians are the third and most abominable of all’.

Source: Unwell

Thus, despite the Greek cultural influence, Carians were not perceived as Greeks. Clearly, there was a limit to this Greek influence or else it seems unlikely that they would have been held in such low esteem by the Greeks, especially Ionians.

So, who else might have influenced Carian attitudes towards sibling marriages? Simon Hornblow, in Mausolos (1982), notes that there are "confirmed" examples of sibling marriages in Bactria, Pontos and Parthia (among other "sub-Iranian cultures". He also notes the historic connections between Egypt and Caria (e.g. Carian mercenaries being sent to Egypt) and the influence of the Pyramids on the Mausoleum at Halicarnassus. However, it is impossible to assert that all, or even any, of these factors had any influence on how Carians felt about sibling marriages.

Lastly, it is worth considering attitudes to royal sibling marriages in other cultures:

Royal incest, notes historian Joanne Carando, was "not only accepted but even encouraged" in Hawaii as an exclusive royal privilege.

In fact, while virtually every culture in recorded history has held sibling or parent-child couplings taboo, royalty have been exempted in many societies, including ancient Egypt, Inca Peru, and, at times, Central Africa, Mexico, and Thailand.

Three other points worth noting are:

1. The Hecatomnid sibling marriages appear to have happened only in that one generation. They should thus probably be seen as special cases.
2. Neither of these sibling marriages are known to have produced any children (whereas the third son Pixodarus did have a daughter - her Persian husband, Orontobates, was appointed by the Great King as successor to Pixodarus). Whether the lack of children from the sibling marriages was intentional, or simply because they had children who died young, meaning all trace of them has disappeared, is unknown.
3. We do not know if the siblings had the same mother or different mothers, nor whether this would have had any relevance.