I have the same question on how to record these locations, and I haven't come to a decision. I've leaned towards using the current name (because that's who now has the records), and including the original name as an "alternate fact".

I like using Google's ability to suggest alternate search terms to help resolve misspellings and help me track down the "real" names of places. Of course, it also suggests links to modern municipal websites and wikipedia pages, which are sometimes helpful on teasing out the tangles of geographic history. Google Earth is also nice.

In my case, my wife's ggparents listed their country of origin on various documents as Poland, Austria, and "Galicia". Another issue is that (I believe) on US Censuses, the census taker would try to record the name of the country at the time of the census: a lot changed in eastern Europe between 1880 and 1950, and they may not have kept track! A person may have emigrated from an Austrian village in 1900 may have listed as Poland on a 1930 US Census. On top of that, common folk may honestly not know what country has control of their home town which is half-a-world and several decades away. Another funny case: another set of ggparents appears on US Censuses as coming from Syria, but the villages they came from are now in eastern Lebanon. When Ancestry.com sees "Syria", it wants to update it to the modern "Syrian Arab Republic", which is the current name of the country, but their home towns were never part of the SAR. When they left, it was the Ottoman Empire; then it was the French Syrian Mandate; then Lebanon. In this case, I think "Lebanon" is the only sensible choice for recording the location. Luckily, the city names haven't changed, so are unambiguous.