Well, this thing is certainly taking on a life of its own! I had originally thought to do the major individual awards in one go, but there are just too many of them, so I'm breaking them into two parts. I've decided to discuss the awards in chronological order of inception, and, just because I can, I'll be mentioning any members of the Oilies who have held these awards.
Sources for this are NHL.com, Hockey Reference, the Hockey Hall of Fame, the International Jewish Sports Hall of Fame, and Wikipedia.
MAJOR INDIVIDUAL TROPHIES, SEGMENT ALPHA:
The Hart Memorial Trophy: Awarded to the NHL's Most Valuable Player, the Hart Memorial trophy is not, in fact, the oldest individual trophy in the league, as it was first awarded in 1960. However, I've decided to count it as such since it represents a continuation of the original Hart trophy, which was donated way back in 1923 by Dr. David A. Hart. Dr. Hart was the father of Cecil Hart, who coached the Montreal Canadiens for 9 seasons between 1926 and 1939, winning 2 Stanley Cups in the process, and is one of a smallish number of Canadians to be inducted into the International Jewish Sports Hall of Fame. There's a small measure of historical irony in the fact that Cecil Hart was a direct descendant of Aaron Moses Hart, an officer in the British army that took Montreal from the French in 1760.
Wayne Gretzky (8 times) and Mark Messier have won the Hart while members of the Edmonton Oilers.
The Lady Byng Memorial Trophy: Awarded to "the player adjudged to have exhibited the best type of sportsmanship and gentlemanly conduct combined with a high standard of playing ability." I believe that the NHL is unique among major North American professional sports leagues in having an annual award for sportsmanship. Anyway, the trophy was donated to the league in 1925 by Marie Evelyn Moreton, aka Lady Byng, the Viceregal Consort of Canada (i.e the spouse of the Governor General) between 1921 and 1926 (the "Memorial" bit was added after her death). She was married to Julian Byng, 1st Viscount Byng of Vimy, who had commanded the Canadian Corps for a time during the First World War, most famously at Vimy Ridge. During his subsequent Governor Generalship, the Byngs were frequently to be spotted at Ottawa Senators games, and the donation of the trophy followed from that. After Viscount Byng's death in 1935, Lady Byng returned to Canada, and lived the rest of the her life in Ottawa. Sadly, by the time she returned, the Senators had folded.
The Oilers have had two winners of the Lady Byng Trophy, namely Wayne Gretzky and Jari Kurri.
The Vezina Trophy: This trophy is now awarded to the league's best goaltender, as voted upon by NHL general managers (it used to be awarded to the goalie or goalies whose team allowed the fewest goals in the season). It is named, of course, after the legendary Georges Vezina, nicknamed "le Concombre de Chicoutimi" for his coolness in net, who tended goal for the Montreal Canadiens from 1910 until November of 1925, when he was forced to retire by the tuberculosis which would kill him a mere four months later. In the wake of his death, the Canadiens donated the Vezina trophy in his honour, and it was first awarded after the 1926-27 season (the first winner of the trophy was, in fact, George Hainsworth, Vezina's successor in the Canadiens' goal). There are a number of myths out there about Vezina, including that he didn't learn to skate until he was in his late teens (true), and that he fathered 24 children, 22 of whom died in infancy (false - Vezina had two sons).
Grant Fuhr is the only Oiler to have won the Vezina.
The Calder Memorial Trophy: The wording for this one is quite elegant as well; it is awarded to "the player selected as the most proficient in his first year of competition in the National Hockey League." The trophy is named after Frank Calder, who never played hockey but was instrumental in the creation of the NHL and was the league's first president. He presided, ruthlessly, over the league's early growth and expansion to the U.S., and its subsequent emergence as the top professional hockey league on the continent. Calder also, in 1933, initiated the practise of rewarding each season's top rookie, although the trophy itself was formally inaugurated until after his death in 1943. As a final historical note, Calder was staunchly opposed to segregation in sports. He is reported to have said, at one point in the late '20s, "[p]ro hockey has no ruling against the colored man, nor is it likely to ever draw the line."
No Oiler has ever won the Calder trophy. However, Oiler fans will swear up and down that Wayne Gretzky was unjustly denied the award in 1979-80, when the NHL decided that Gretzky's brief pro experience in the WHA rendered him ineligible. Of course, that didn't prevent Sergei Makarov, who had a mere 13 seasons' experience in the top Soviet league, winning it in 1990. To its credit, the league has since closed that loophole.
The Art Ross Trophy: Awarded to the league's top point-scorer in each season. The trophy was donated to the NHL in 1947 by its namesake himself. Ross, who ironically was a defenceman during a hall-of-fame playing career in the pre-NHL days (Ross may have been hockey's original offensive d-man, a sort of early version of Bobby Orr), is probably best known as the long-time coach of the Boston Bruins, a position he occupied for 16 seasons between 1924 and 1945. He is actually credited with having chosen the name "Bruins." He also managed the Boston club from 1924 until 1954. Ross had reputation, throughout his involvement in hockey, for belligerence; while playing for the Ottawa Senators in 1915, he got into a fight with Toronto's Minnie McGiffin that ended with both players spending the night in jail.
Seven of Wayne Gretzky's ten Art Ross Trophy wins occurred while he was a member of Oilers. No other Oiler has ever won it.
The James Norris Memorial Trophy: Awarded to the league's top defenceman. the trophy is named after James Norris Sr., and was donated to the league by his children in 1953. The Norris dynasty, which accumulated vast wealth in grain and cattle in the 19th and early 20th centuries (it is hard not to imagine the word "baron" in connection with the Norrises) owned the Detroit Red Wings from the early '30s until Bruce Norris sold the team in 1982. James Sr. also had significant ownership stakes in the Rangers and Black Hawks at the same time as he owned the Red Wings, which you probably wouldn't be able to get away with today. And, as you can see from the picture above, he has a ship named after him!
Paul Coffey won the Norris trophy twice for the Oilers.
The Conn Smythe Trophy: This one's awarded to the most valuable player in the Stanley Cup playoffs, and was presented to the league by Maple Leaf Gardens Inc. in 1964. It is named after Constantine Falkland Cary Smythe, owner of the Toronto Maple Leafs from 1927 to 1961. Smythe was no hands-off owner - he also managed and briefly coached the Leafs, and financed the construction of Maple Leaf Gardens. He also saw action in both World Wars, winning a Military Cross and doing time as a POW during the First, and being horribly wounded during the Second. And he is of course remembered for his famous summation of his hockey philosophy: "If you can't beat 'em in the alley you can't beat 'em on the ice."
Three Oilers have won the Conn Smythe Trophy: Gretzky (twice), Messier, and Bill Ranford.
Well, that's about half of the major individual trophies. Next time up: the other half! Stay tuned to find out which is the only NHL trophy to be named after a Nobel laureate!