Neil Winskill's side eased past Leeds United in the fourth round following a 4-1 victory at Whitley Park and have been rewarded with a home clash against the Hornets (kick-off 2pm BST), with the cup tie taking place at St. James' Park.
Stephenson, who scored a first-half hat-trick against the Whites last week, is hopeful of adding to his goalscoring tally whilst relishing the prospect of playing on the hallowed Tyneside turf for the first time.
The 18-year-old forward, a Newcastle United season ticket holder growing up, told nufc.co.uk: "Everyone's buzzing for it. We haven't been in the fifth round for quite a while now so everyone is really up for it and excited to play, especially at St. James' Park.
"I'm looking forward to playing in the stadium. It would be a proud moment for me as every Newcastle fan dreams of playing at St. James' Park. It would be a privilege to get the chance to play there, in such a big game, for the club.
"It is a mint game against Leeds. Everyone played really well and I was really happy to score a hat-trick. It was great to help the team, get the win and progress into the next round.
"I've spoken to Neil Winskill about scoring more goals and people recognising that. To go and score a first-half hat-trick in the Youth Cup was really good and it has given me a big boost in confidence."
The young Magpies bid to reach the quarter-finals of the prestigious tournament for the first time since 2017 and, should they advance past Watford, will face another home tie against either Aston Villa or Burton Albion in the last eight.
Stephenson, who made his debut for United's under-23 squad at the age of 16, is focussed on playing the game rather than occasion against Watford, who progressed into the fifth round following a 3-2 away win at Lincoln City after extra-time.
"Everyone needs to be switched on, be ready from the first minute, and not think about where we are playing," the Blyth-born forward added. "Some of the players in the squad have played at St. James' before so they already have that experience of playing in such a big stadium.
"All the lads are just wanting to win to reach the next round and possibly play another game at St. James'.
"The cup games and the derbies against Sunderland are the ones we always look out at the start of the season and since we've got quite far in the cup at the minute, I think this should be our main focus.
"It would be an amazing feeling for the players, staff and even the fans if we can go on a good run. If we are all switched on and have our game heads on, we have a squad which are more than capable of progressing further in the competition."
