Dictionary.com has etymology.

In a nutshell, there is no connection between the words.

"Lad" is from around 1300 AD. Lady is from before 900 AD.

Lad

c.1300, ladde "foot soldier," also "young male servant" (attested as a surname from c.1100), possibly from a Scand. language (cf. Norw. -ladd, in compounds for "young man"), but of obscure origin in any case. OED hazards a guess on M.E. ladde, plural of the pp. of lead (v.), thus "one who is led" (by a lord). Meaning "boy, youth, young man" is from c.1440. Scottish form laddie, a term of endearment, attested from 1546.

Lady

Origin:
bef. 900; ME ladi(e), earlier lavedi, OE hlǣfdīge, hlǣfdige, perh. orig. meaning “loaf-kneader,” equiv. to hlāf loaf + -dīge, -dige, var. of dǣge kneader (see dough; cf. ON deigja maid); see lord

ady
M.E. lafdi, lavede, ladi, from O.E. hlæfdige "mistress of a household, wife of a lord," lit. "one who kneads bread," from hlaf "bread" (see loaf) + -dige "maid," related to dæge "maker of dough" (see dey (1); also compare lord). Not found outside English except where borrowed from it. Sense of "woman of superior position in society" is c.1200; "woman whose manners and sensibilities befit her for high rank in society" is from 1861 (ladylike in this sense is from 1580s). Meaning "woman as an object of chivalrous love" is from late 14c. Used commonly as an address to any woman since 1890s. Applied in O.E. to the Holy Virgin, hence many extended usages in plant names, etc., from gen. sing. hlæfdigan, which in M.E. merged with the nominative, so that lady- often represents (Our) Lady's; e.g. ladybug. Ladies' man first recorded 1784.