Re: the Ark of the Covenant,
Tudor Parfitt, one of these 'real' Indiana Joneses we hear so much about, reckons he's found it - or a bit of it - unlabelled in a Zimbabwean museum. His claims, so far as I've followed them (not having read the book, but I've heard him interviewed and read articles) are intriguing - inconclusive, at the end of the day, but maybe the case he's put together is as close as we'd ever get... certainly closer than I'd have expected anyone to get.
Personally, I'd like to see anything,
anything, directly connected to Jesus. The closest things so far are what
may be
John the Baptist's Cave, and two fair contenders to be the
remains of St Peter. Beyond this there's very little I'm aware of.
Also rather good would be any NT manuscripts, however fragmentary, which could help to confirm or reject current theories as to the date and authorship of the Gospels.
I'm of the opinion really that OT artifacts will be very difficult to find, since the places, names, dates and events described in those ancient books have arrived with us through so many cultural and mythological filters. So many of the tales either cannot have happened, or have been recorded in older versions with entirely different details in Canaanite, Babylonian, etc. mythohistoriography that to suddenly identify an artifact with, say, Abraham, might well be like suddenly claiming to have dug up Uncle Sam's hat in the White House garden.