Botham, 18, is the grandson of legendary England cricketer Ian Botham and son of former Cardiff wing Liam.
Frewen, 29, plays for Newport RFC and impressed for Newport Gwent Dragons regional sevens team last summer.
Coach Gareth Williams said they came in with an eye on the 2018 Commonwealth Games and World Cup Sevens.
Wales have reached the semi-finals and quarter-finals of the first two rounds of 2016-17.
Botham is in Cardiff Blues' academy and played for Wales Under-18, scoring on his debut in 31-10 win over their Scottish counterparts in March, 2016.
"James Botham's selection supports the work we're trying to do in developing players with regions," said Williams.
"He's a very promising player for Cardiff Blues. He's trained hard and has earned his opportunity in Wellington.
"Elliot has got a really good pedigree on the invitational sevens series, and he's been part of the development group [assistant coach] Richie Pugh set up at start of season. It's a fantastic opportunity for him."
Wales are led by GB Rio Olympics Sevens silver medallist Sam Cross.
Wales face Scotland, Canada and Russia in Pool D in Wellington.
Wales Sevens squad for Wellington Sevens: Owen Lane (Cardiff Blues), Morgan Williams (Ospreys), Steffan Andrews (Aberavon), Elliot Frewen (Newport), Owen Jenkins, Tom Williams (Ospreys), Ethan Davies, Lloyd Evans, Adam Thomas, Sam Cross (capt), James Botham (Cardiff Blues), Ben Roach, James Benjamin (Newport Gwent Dragons), Matthew Owen (Scarlets).
Teenager James Botham and Elliot Frewen are set to make Wales Sevens debuts in the World Rugby series in Wellington between 28-29 January.