The club were promoted to the National League after winning the North Promotion Final against Brackley back in May.
Some impressive signings has raised hopes that they can have a good debut year in Non-League football’s top flight.
Weaver has given an interview with the Harrogate Advertiser in the build up to Saturday’s season opener at home to Sutton United.
He told them just avoiding relegation isn’t quite enough for him or his squad.
“I think that at boardroom level, survival would probably be deemed acceptable as well, but within the dressing room we have our own targets,” he told them.
“We want to do more than just stay up. We are determined to make an impression and won’t class survival as a success.
“We’ll try our hardest to ensure that we are in the top half of the table, and although I’m not saying that we must do that.”
Among Harrogate’s summer signings are Brackley top scorer Aaron Williams, defender Callum Howe and forward Jack Muldoon.
Weaver added: “I’d love to get everyone excited again by having a team that still has something to play for and is chasing a play-off place come the last few months of the campaign.”