Is the etymology of the name Yudhishthira correct?


Etymology of Yudhishtira = steady in war, is correct. Here, the war can be any kind of battle or struggle. Throughout the Mahabharata, Yudhishtira was known to behold the Dharma & remaining steadfast in every challenging situations.

The passage you described about Yudhishtira 'wanting' to go in exile - is just part of warring frustrations. That should not be considered as a proof, as it really didn't happen.
There is another example (read last passage in the link), when frustrated with Drona's overwhelming power & Arjuna's inability to stop him, Yudhishthira taunted Arjuna about forfeiting Gandiva bow & hearing that Arjuna wanted to literally kill his elder brother to fulfil a vow!

Such things are usual during any war. Similar exile sentiments were echoed by Arjuna during Bhagavad Gita's discourse as well. Bhima also had shown such passive sentiments against the war itself, even before it had begun. See Bhima's view of Kurukshetra war.

How was he named "Yudhishthira"?

Kunti begot a child by Niyoga with the god of justice (Dharma-rAjA).

And as soon as the child was born, an incorporeal voice (from the skies) said, 'This child shall be the best of men, the foremost of those that are virtuous. Endued with great prowess and truthful in speech, he shall certainly be the ruler of the earth. And this first child of Pandu shall be known by the name of Yudhishthira. Possessed of prowess and honesty of disposition, he shall be a famous king, known throughout the three worlds. [Adi Parva]

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