David Ortiz reached base 297 times in 2007, the most times any active Red Sox player has reached base in a season.

Howard Ehmke allowed 457 baserunners in 316.2 innings in 1923, the most baserunners allowed by a Red Sox pitcher in a single season.

Red Ruffing lost 25 games in 1928. He finished the season 10-25, and his losses are the most by any hall of famer who played for the Red Sox. 

Johnny Peacock had 39 RBI in 195 at bats in 1938, the highest season total for any Red Sox player with less than 200 at bats.

Don Baylor was hit by a pitch 34 times in 1986, and 24 times the following year. No other Red Sox player has been hit more than 17 times in a season. 

Carl Yastrzemski was 44.041 years old on 10/2/1983, the oldest player ever to play for the Red Sox. Rickey Henderson was the second oldest player, at 43.278 years.

Jim Rice caused 509 outs in 1984. Outs are calculated as (AB-H+CS+GDP). Rice grounded into 36 double plays that year. 

Wade Boggs owns the Red Sox record for the most plate appearances in a single season. He had 758 in 1985, while batting .368 with a .450 OBP and .478 SLG. 

Calvin Pickering played 17 games in 2001 with a listed weight of 275 pounds. He shattered the old weight record of 250 that was set by Rich Garces and Sam Horn. 

The Red Sox did not ground into a double play from 4/12/2000-4/25/2000, a span of 10 games.

Duffy Lewis was caught stealing 31 times in 53 attempts in 1914. That same year, teammate Tris Speaker was got caught 29 times, Larry Gardner 23 times, Hal Janvrin 20 times, and Harry Hooper and Everett Scott were caught 14 times each. 

Roger Clemens owns the Red Sox record for the most quality starts in a season. He had 27 in 1990. 

Nomar Garciaparra had a hit in 30 straight games from 7/26/1997 - 8/29/1997. He had a .383 BA and 1.059 OPS during that span. 

Mark Bellhorn struck out 177 times in 2004, shattering the old Red Sox record of 162 set by Butch Hobson in 1977.

The Red Sox sent 84 batters to the plate on 9/3/1981, a team record for plate appearances. They had 23 hits and 7 bases on balls, but only managed 7 runs in a 7-8 loss to the Mariners.

Oldest player to have a season OPS above 1.000

In 1960, Ted Williams posted an OPS of 1.096. He was 41 years old.

note: For the sake of sample size, I only included batters with more than 200 at bats in my query. Williams had 310.