Actually what he was offered was vinegar. Not a pain killer, (Rome wanted their executed to suffer) nor a poison. (Rome wanted to stretch it out as long as possible)
There appears to be some variation, but the combination of words used in Matthew and Mark describe pain killing properties within the range of their definitions.
[ Making use of KJV + Strong's Greek and Hebrew Dictionaries ]
They gave him vinegar3690 to drink mingled with gall:5521 and when he had tasted thereof, he would not drink. <---> Matthew 27:34
G3690
ὄξος
oxos
ox'-os
From G3691; vinegar, that is, sour wine: - vinegar.
G5521
χολή
cholē
khol-ay'
Feminine of an equivalent perhaps akin to the same as G5514 (from the greenish hue); “gall” or bile, that is, (by analogy) poison or an anodyne* (wormwood, poppy, etc.): - gall.
*anodyne - capable of relieving pain.
And they gave him to drink wine3631 mingled with myrrh:4669 but hereceived it not. <---> Mark 15:23
G3631
οἶνος
oinos
oy'-nos
A primary word (or perhaps of Hebrew origin [H3196]); “wine” (literally or figuratively): - wine.
G4669
σμυρνίζω
smurnizō
smoor-nid'-zo
From G4667; to tincture with myrrh, that is, embitter (as a narcotic*): - mingle with myrrh.
*narcotic - a drug which causes stupor or numbness.
Also in Pslams -
They gave me also gall7219 for my meat; and in my thirst they gave me vinegar2558 to drink. <---> Psalm 69:21
H7219
רושׁ ראשׁ
rô'sh rôsh
roshe, roshe
Apparently the same as H7218; a poisonous plant, probably the poppy* (from its conspicuous head); generally poison (even of serpents): - gall, hemlock, posion, venom.
*Poppy - plant from which opiates are derived.
H2558
חמץ
chômets
kho'-mets
From H2566; vinegar: - vinegar.
The following account gives credence to the idea that the 'drink' offered to Christ was to 'quiet him', from crying out in agony, or in other words 'to ease his suffering'.
And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?
Some of them that stood there, when they heard that, said, This man calleth for Elias.
And straightway one of them ran, and took a spunge, and filled it with vinegar, and put it on a reed, and gave him to drink.
The rest said, Let be, let us see whether Elias will come to save him. <---> Matthew 27:46-49
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The following is a quote from a modern day herbalist -
Vinegar has been used for thousands of years to preserve food, disinfect surfaces, deodorize and make medicinal formulas. In fact, before stills were used to produce high-proof alcohol, most herbal extracts were made with solvents like water, wine and vinegar.
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The people who offered Christ the fortified vinegar are referred to as 'they'. They could of been some of the roman soldiers, and / or some of the Jews who came along with them. Either way it likely tasted bad / bitter, and if someone were to drink it fast, (thinking it was water to quench their thrist), it would likely gag them...so there may well have been some 'cruel intentions' mixed in, (as is implied in Psalm 69 ), along with those who likely sought to offer a pain reducing formula. (When he cried out in agony.)
