I always thought Archaeology was a very cool profession, and it's a shame that they abandoned it in retail. Classic purists might not be on board with it since it was added in Cataclysm, but I would love to see the profession revived for Classic+. Maybe with a renewed focus on lore and exploration rather than collecting a ton of junk, similar to getting all the runes in SoD.
This is honestly a great idea. Reduce the tediousness of finding the digsites a tad maybe, and add in some cool lore stuff. Imagine finding an enchanted arrowhead from the War of the Ancients or something like that? Could add incentive by having you trade this stuff in for a currency you can use to buy things like mounts/pets from a vendor.
I hated Archaeology with a passion in the original WoW run. But during Cata Classic I gave it a shot and found I was enjoying it more. So I wouldn't be opposed. It's funny how in the lore as the expansions went on, dwarf archaeologists became more prominent, while the profession was removed.
Yeah, that's a good one. The three searches per dig get old real fast.
They have some neat puzzle mechanics they use for daily quests in retail now, so maybe something like that could make it more interesting than the old digsite system. But there I go suggesting more retail stuff 
Oh yeah, any excuse to let them use more of that Bejeweled mini-game they had in BFA. That thing got me to reinstall Puzzle Quest.
