From the Weidlich book, I would recommend the following as 'easiest':
De Bone in de Barnyard- sort of like an exercise disguised as a tune, in that mastering its simple little phrases of fingering will help you in future tunes.
Lucy Neal is pretty easy.
Mary Blane and Old Dan Tucker are not too hard, and both get you familiar with doing the broad 'back and forth' brush with using your thumb on the second half of the brush. Important. I would say Dance, Boatman, Dance falls into this category too, but its timing is just a bit more complex.
Old Dinah's Goin to Town is like a simple exercise.
Old Johnny Boker- definitely, and it's very popular in jams.
Walk Along john - very easy. I like it a lot despite its total simplicity. The fingering in the 3rd-to-last measure is essential to learn for playing other tunes.
Many would say Camptown Hornpipe but just be sure to get the timing of the critical dotted note right. Corn Shucking jig is a basic essential tune but it might be easier after you get the above tunes down.
When I was first starting out I found Old Dan Tucker to be fairly easy. I knew the tune from playing it on my dulcimer. I also liked Briggs Corn Shucking Jig, Briggs' Reel and Circus Jig. I found these in the Briggs 1855 book, I haven't seen the Wiedlichs version though.
Dandy Jim came easily to me. For some reason the melody stuck in my head and I was able to plunk it out by ear in a few minutes. Embellishments came latter.