A subsequent spell at White Hart Lane during 2001-04 was blighted by injuries but the South American still scored 23 goals in 98 games for Spurs.
"The Tottenham days I remember the most because I was older but the [2000 FA] Cup final really stands out as well. I went on the pitch afterwards and I remember I was quite scared. I was very young and surrounded by so many people," said Poyet, who was 5 at the time.
Having a famous father is always a double-edged sword - it can provide opportunities but also inevitable comparisons. Poyet senior, who went on to manage Brighton and Sunderland, has cast a giant shadow.
"It's been hard sometimes - people remember him, like to talk about him," said Poyet, who admits some people have called him Gus.
"It just slips out. They start calling me Gus. They are used to seeing the surname and to seeing my dad's name next to it. But when they see me play, they realise we are different players. If we played the same style, the same position, you might have everyone comparing us 24/7."
At this early stage, Poyet has done well to create his own niche. The defensive midfielder was named Player of the Year for the 2013-14 season at Championship club Charlton when aged 19, before securing a four-year deal at West Ham. He made only four first-team appearances for the Hammers this season but is prepared to bide his time.
"We signed a lot of players, which pushed me back a bit, said Poyet. "But I was taking a huge step from Charlton to West Ham and I knew the first season would be tough. Hopefully after this World Cup I can push on."
Poyet has only recently resolved his international allegiances, opting for La Celeste after representing England at under-16 and under-17 level.
"It was tough because I have lived in England for 16 years," he said. "But I had to choose and, in my heart and with my family, it was Uruguay."
Talking with Poyet is an unusual experience - a Uruguayan footballer with a Cockney accent. But his fluency in English has been appreciated by his team-mates in New Zealand.
"Every time someone needs something here, they ask me," laughed Poyet. "At lunch, at dinner, meeting people ... I had to take someone to the hairdressers [last week] because they didn't know what sort of haircut they wanted. It's crazy for me but I try to help as much as I can."
Uruguay, runners-up at the 2013 Under-20 World Cup, have enjoyed a low-key preparation, training in the picturesque surroundings of the Hibiscus Coast Football Club in Stanmore Bay. They've drawn big crowds every day from the local community and enjoyed a stirring kapa haka performance by a local school group on Friday.
"I'm used to the rugby one but it was very impressive," said Poyet. "For such young kids, the passion they put into it, it was quite fierce. Some of the boys didn't know what it was but they were impressed."
Uruguay travel to Dunedin today ahead of their first match against Serbia next Sunday. They face Mexico on June 2 before their final group game against Mali in Hamilton on June 6.
