I'll take a shot at it and I'll give it a shot:
slang words for "effort" in the history of American English

(Based on Oxford Dictionary of Slang (Ayto), pp. 246-47)

Some fun pictures

 

(Sorted by COHA)

GB

COHA

COCA

First occurrence (COHA)
(with previous literal meaning)

put *self out

26,932

121

129

1838. I rather put myself out of the way for it

bend over backwards

7,841

68

246

1951. In fact, I bend over backwards to see that they get every customary attention
1931. he drew away, bent over backwards till the stick was touching the ground

take some doing

8,095

60

66

1935. Of course that took some doing; it took a good deal of doing

knuckle down

5,987

54

20

1869. who are invariably ordered to " knuckle down and bore it hard,
1832. If they had separated in pairs, and knuckled down at marbles on the side-walks

pull out all the stops

9,437

50

164

1938. pulling out all the stops in tearful eulogies to the Union dead
1908. and seating herself at the organ, pulled out all the stops

go for broke

5,282

33

102

1943. I have just spent a week with the 442nd (Go for Broke) Combat Team at Camp Shelby, Mississippi

beaver away

134

9

12

1979. even now beavering away on a screenplay version of her best-selling assault

have/take a NOUN at (it / vvg*)

shot

27,256

248

516

1897. Suppose you take a shot at it, and see what you can do
1920. But once she took a shot at living here, she ought to stick by it

crack

9,354

130

108

1935. Say, everybody around here has taken a crack at it one time or another
1935. might ... take a crack at carpentering

go

14,587

84

79

1894. I'm going to have a go at it, anyway!
1960
. After that, we could have a go at exchanging news

stab

4,534

27

106

1959. No one knows, but Novick is willing to take a stab at it.
1982
. perhaps new creatures will have a stab at fulfilling His will

whack

2,181

23

27

1869. Well, I'll have a whack at that room myself
(no V-ing)


Are "he had/took a crack at it" the same thing?

take a NOUN at (it), take a NOUN at V-ing

have/take a [nn1] at [vvg*] : COHA, COCA

turn 1852, hand 1868, (chance 1905), shot 1920, crack 1935, fling 1956, stab 1988, swing 1993, whack 1998, try 1985
HAVE: knack
No V-ing: go

Why these nouns?
Which are the first ones? (turn, hand, chance)
How did it spread? Why? (shot, crack, fling, stab, whack in 1900s)
Future extension: swipe, swat, punch, slap ??
 

give it a NOUN

       

whirl

2,232

28

42

1911. I'll make it right with you. " " Seein' it's you, " says I, " I'll give it a whirl.
1905. put her small hand to the wheel of fortune and given it a whirl
1845. thick rich fluid fell back into the kettle, he suddenly gave it a whirl


give it a [nn1] : COHA, TIME, COCA
Earlier vs later
trial 1845, try 1850, chance (1876), shot (1863, 1904, then all later), whirl (1845/1911), go (1941), test (1962, 1989), start (1902, 1993), whack (1996), (spin), (look 1993)
give it a rest / break ( = "stop trying")