The youngest that I can find evidence of was Irving Hanchett, executed in Florida at 15 on 5/6/1910.
Irving Hanchett, barely 15 years of age, was executed by Florida in
1910. Only three months elapsed between his crime (the murder of a teenaged girl who rejected his advances) and his execution. Hanchett
had just moved to Florida from Connecticut and had no frieds or family
in the area. While he was awaiting execution, a priest baptized him
into the Catholic faith, and Hanchett seemed to be buoyed by the hope
of salvation. As the 15-year-old mounted the scaffold, he said:
"Marcy, my Jesus, my Jesus, mercy. Goodbye everyone" (Florida Times
Union [Jacksonville], 7 May 1910).1
Note that there are some discrepancies as to his age, for example this newspaper article reports him as being 18:
Deland, Fla., April 8. -- Irvin Hanchett, the 18-year-old boy charged
with the murder of Clevie Tedder, aged 14, on February 12, was placed
on trial here today. The child's body was found near the roadside,
not far from her home, with more than sixty knife wounds, made with an
ordinary pocket knife.2
However, the preponderance of materials I could find gave his age at 15.
Note that this is the youngest that I can find substantiating record of. The database at deathpenaltyusa.org lists an unnamed white male executed for arson in Suffolk County, Massachusetts in 1837 at the age of 13, but I can't find any source material that would corroborate that.
In response to the edited question, the youngest person (and certainly the youngest female) generally believed to have been legally executed in the United States is Hannah Ocuish, executed at age 12 in Connecticut in 1786.
Note that the deathpenaltyusa.org site linked about records 2 other 12 year olds executed, both black slaves, a boy in Virginia by the name of Clem on 5/11/1787 and another boy named Bill on 7/30/1791. In the latter case, there seems to be a disagreement on the location. I'm guessing that the LA Times is the more accurate account.3
1 Radelet, Michael ed. Facing the Death Penalty: Essays on a Cruel and Unusual Punishment, p 52
2 Morning Examiner. (Bartlesville, Okla.), Vol. 15, No. 46, Ed. 1 Saturday, April 9, 1910, p 3
3 Los Angeles Times. Death for Juvenile Crimes : Execution, a Practice Dating to 1642, May Continue This Week, January 07, 1986 p online