2008 really grew on me. No expectations and it felt like the team was starting to get hot at the right time. Just ran out of games. Obviously the offense was significantly worse without the GOAT and injuries didn't help but they did play some good football at the end and if they got in the playoffs who knows?
2008 was the first that immediately came to mind, for the same reasons.
After an incredibly stressful 2007 when the Pats were universally painted as the big, bad villains, it was enjoyable to be the underdog winning against all odds the following season.
1974
After seven consecutive losing seasons the Patriots jumped out to a 6-1 start. Out of nowhere the Pats were on a tear, beating the Colts 42-3 and Jets 24-0 in consecutive weeks. It culminated in beating the NFC's best team, Minnesota, in the Bob Windsor Game. Unfortunately injuries took their toll the rest of the way - but there was hope after so much bad play in the Clive Rush-John Mazur era.
1966
The two leagues were scheduled to play in the first "AFL-NFL Championship Game", something that was later known as Super Bowl I. Jim Nance was the AFL MVP and led the league in rushing. The Chiefs were going to win the AFL West, and Pats fans were looking forward to a rematch; the Patriots had played well in a game in KC a few weeks earlier that ended in a tie. All that was needed was a win over the lowly Jets in the final game of the regular season ... then beat KC and the Patriots would represent the AFL in the historic game. But no, the team was inexplicably flat, turned the ball over three times, was unable to stop the Jete running backs while young Joe Namath threw three TDs - and Buffalo passed them in the standings to win the east by a half a game. Incredibly disappointing finish, but the rest of the season was a lot of fun to watch.
1991
Coming on the heels of the 1990 1-15 Rod Rust/Zeke Mowatt/Lisa Olson season, 1991 was a breath of fresh air. The Pats won five more games than in the previous season, and while the offense still sputtered, the defense improved from atrocious to at least middle of the pack. It was socially acceptable to watch and root for the Patriots again.
1993
It didn't begin well at all - the Pats began the season 1-11 - but the last four games, all wins, gave genuine hope for the future. Winning the final game of the season over the Dolphins in OT - thus denying Don Shula a playoff berth - was enormously satisfying.
1964
It's so long ago and I was so young, I don't remember any specifics - other than that was when I first became a fan of the Pats. They had a good team, which was helpful as a very young fan, going 10-3-1 - the best record in the first 15 years of the franchise. Caught any game that was televised.