In Mary's personal letters....

[RJNorton]

In Lincoln's Sons Ruth Painter Randall wrote, "Dear little Taddie was named, for my husband's father, Thomas Lincoln - no T - for a middle name - was nicknamed, Taddie, by his loving Father," wrote Mrs. Lincoln a dozen years after his birth. That seemed fair enough - to name one son for the grandfather on the Todd side and one for the grandfather on the Lincoln side. The baby's head was larger than usual and Mr. Lincoln, viewing the contrast between head and tiny baby figure (perhaps looking on as his mother gave him a bath), called him a little Tadpole, and thus started a nickname that was ultimately to be known all over the nation."

I have also seen the "as wriggly as a tadpole" phrase used for Tad's name, but I do not know the original source for that. I also do not know the source for Ruth Painter Randall's version.
We have an original source (Mary's letter) that it was Abraham who gave Tad the nickname, but at this point, I have not yet seen an original source for exactly why the name was given.