My favorite lost causes (in no particular order) are:
A more tolerant pagan-Christian-Jewish mode of coexistence developing in Antiquity (frex the Valentinianic model, Julian's state, or a world where Christianity develops into a following of theosebei - not fully converted Gentile synagogue hangers-on - and never catches on in maionstrweam paganism).
An printing-induiced 'value survival' at a point in history where a multipolar, religiously tolerant society is at its apogee (like in the pre-Seljuk days of Islam, Sung China, or the later Hellenistic or Roman eras, or maybe in India, though that society may not have been literate enough). Europe has had a rough time because its 'value survival point' is rooted at a time when division, conflict, and segregation were all the rage and purity counted.
Chinese or Indian development to rival Europe's. By 1700, China still had an economic output and productivoity second to none. 100 years later, it was falling back. What if not? (Yes, I know, Cheng He is sooooo 70s. I still like it)
The Revolutions of '30 and '48.
the Weimar Republic. I like Weimar Germany. I wish there had been some productive way of dealing with all that centrifugal force. For all the cozy consensus-based comforts of the FRG, it gets stuffy here.
The post-Assyrian Aramaic city culture of the Near East. What a cultural model to play with. Imagine if we had more of its literature (basically, we have the second-temple era books of Bible and what fragments survive in other cultures).
Hindu India. Mughal India, too, but to a lesser extent.
Socialism. I mean, real Socialism, not the Marxist-Leninist-Stalinist-Maoist nightmare it became after the doctrine of revolution as the only way split the movement.
A medieval Italian kingdom. Preferably based on a pre-Papal Lombard tradition.
The world of the mid- to later 19th century. I know you can't preserve a mindset. But sometimes I wish you could.