Drinking game criteria dictates using a word with frequency near the target number of drinks, which depends on drink concentration. In Hamlet that ranges from "absolute" (2) to "that" (389). If shots, in the 5-7 range would be reasonable adult non-heavy drinking number. Beer in contrast would need a common word. I've played with words in the 6-9 range (such as "wisdom" or "Hamlet") and a problem is they go by fast. So picking a higher frequency word makes sense, expecting to miss some. Conveniently the word "drink" occurs 15 times. This allows missing some, or drinking heavily. Furthermore, they're fairly distributed, culminating at the end with poisoned wine.
I therefore propose "DRINK" as an optimal word on which to drink as a drinking game during Hamlet.
Hamlet word occurrences analysis, 10-20 repeats range, by descending frequency:
20 till
20 poor
20 ophelia
20 o'er
20 means
20 friends
20 england
20 before
19 words
19 without
19 once
19 denmark
19 blood
19 believe
18 there's
18 many
18 keep
18 is't
18 hand
18 father's
18 else
18 about
17 young
17 welcome
17 still
17 noble
17 long
17 little
17 first
17 find
17 farewell
17 cause
16 youth
16 thoughts
16 that's
16 sword
16 speech
16 since
16 question
16 marry
16 majesty
16 hast
16 faith
16 body
15 yourself
15 watch
15 tongue
15 therefore
15 stand
15 said
15 rest
15 players
15 other
15 makes
15 live
15 last
15 drink
15 between
15 better
14 within
14 state
14 spirit
14 something
14 reason
14 murder
14 mark
14 lady
14 heard
14 grace
14 gone
14 gentleman
14 ever
14 dost
14 daughter
14 bear
14 answer
13 word
13 virtue
13 thine
13 sent
13 sense
13 purpose
13 ourselves
13 none
13 haste
13 grief
13 grave
13 goes
13 gertrude
13 friend
13 form
13 duty
13 after
12 while
12 what's
12 we'll
12 voice
12 together
12 thought
12 sleep
12 same
12 revenge
12 remember
12 please
12 norway
12 name
12 myself
12 mind
12 madam
12 hands
12 further
12 fortinbras
12 fire
12 bring
12 being
11 work
11 wind
11 villain
11 twere
11 thousand
11 things
11 stay
11 right
11 peace
11 passion
11 mean
11 kind
11 honest
11 gentlemen
11 full
11 free
11 foul
11 fortune
11 face
11 each
11 e'en
11 doubt
11 deed
11 business
11 best
11 awhile
10 uncle
10 thanks
10 seem
10 second
10 pyrrhus
10 phrase
10 news
10 neither
10 memory
10 let's
10 honour
10 hell
10 he's
10 guildenstern
10 ground
10 given
10 fine
10 excellent
10 every
10 command
10 aught
10 alone
10 alas
Note that this is a simple count and sort by exact word, and does not consolidate variations (such as drinking/drank/drunk). This analysis was based on the same source as my E-Text version, a full classic text but unspecified folio. An interesting observation from this is that Hamlet has over 32700 words, but only about 1700 unique.