Cheapest? Gotta be the .458 Win Mag.
Best? Hard to say...best for what? "Africa" isn't enough of an answer. There are actually several species of critter there, and they're not all the same size.
Seriously, if you have a .375, then any of the .416's just don't seem like that much of a jump, IMHO. If I had no personal history with, and preference for, the .375H&H (which I do) then examining the ballistics would probably lead me to the .416 Ruger as an all-around sort-of-big-bore, that maintains good utility for the smaller African stuff. But with a .375 in your battery, it seems like going to .458 caliber is a logical step. And you'll have a true big-bore!
And, no, Africa is not just for the wealthy. I'm a working man who's been there, and will go back. One of the things that makes this possible is not wasting money. Sure, if funds were unlimited, I'd have a high-grade double rifle in a Nitro Express chambering, but that cost would go a long way towards another hunt, so a .458 WinMag will do the job for a fraction of the cost. Lots of practice is essential, so lower ammunition costs (whether factory or reloads) is important, at least to me.
The ideal choice would be the Lott, but there seem to be many more good used rifles out there in the WinMag. The early problems associated with the shorter cartridge seem to have been ironed out, and I think that the Lott vs. WinMag argument is much like the .338WinMag vs. .340Wby discussion, i.e. not worth worrying about.
JMHO, which is worth what you paid for it and comes with a money-back guarantee.
John