Technics 1200mk2 or 3.
The price went up a bit (lot) since Technics stopped making them, but they are still quite reasonable.
After that, they are insanely reliable and well documented. There is a 78rpm upgrade available (well, there was when they were more readily available - I have not looked in a while) and they come with a very nice arm as long as you don't drop the TT upside-down on it DJ style.
So, yea, 1200.
I would avoid knock offs because they suck.
After that, cheap cartridges are pick your poison.
I am partial to the DL103 as the next step up. Very linear and generally well documented.
For the phono stage, I don't know of anything off the shelf that would work with the budget, but if your friend is DIY handy (like at all) the Hagerman bugle is well regarded for very little money. Look into the "bugle pro" for ideas on how to make the bugle work for 78's
http://www.hagtech.com/bugpro.html
Or just do the de-emphasis digitally.
16RPM is really hard. How many 16RPM discs does he have? Would they be interesting enough to someone else (with a dedicated 16rpm rig) to get them to rip them?