I am leaving for a RTW trip to Argentina/Thailand/Netherlands/Lesotho/Botswana this week and I was wondering if it is recommended to buy an unlocked phone in the US or just wait until I get to Buenos Aires. Will it be cheaper there? Also, does anyone have any recommendations on a cheap phone preferably that does offline maps

I don't know about the other countries, but you are better off buying an unlocked phone in the U.S. instead of Buenos AIres. You will get a better phone for the money, especially if you get a used one from eBay or Craigslist. Plus many of the used phones for sale in BsAs are stolen phones. You can buy a cheap feature phone, but if you want GPS and apps, those handsets will be pricey.
You don't mention what type of phone you want, but I have a Nokia Lumia 620 Windows 8 phone with Nokia maps that works very well without data in BsAs.
However on prepaid phone plans (Movistar or Personal), data (probably 2G) is very cheap, but it is slow. There is a lot of free WiFi in BsAs.

If you get a US unlocked phone, make sure it is quad band. If not it will not work anywhere else in the world.
A phone in BsAs are more sure to be fully unlocked and globally GSM.
You can get a new phone for less that 20USD -- I use a Nokia 1280. good battery life; black and white screen; flashlight. Works everywhere -- except the US and Japan. Chinese phones are even cheaper.