This is the only OT occurrence of the word כָּ֝אֲרִ֗י. Most translations render this word “pierced.” Some see it differently. For example:
Dogs surround me; a pack of evil people circle me like a lion— oh, my poor hands and feet! (CEB)
For dogs are all around me; a company of evildoers encircles me. My hands and feet have shriveled; (NRSV)
Dick Harfield outlines the uncertainty of the Hebrew and its root. Additional information is found in this answer to a related question: How should Psalm 22:16 read?
An obstacle to seeing the word as lion is the use of אַ֝רְיֵ֗ה within the Psalm:
They gape at Me with their mouths, Like a raging and roaring lion (אַ֝רְיֵ֗ה). (v13 NKJV)
Save Me from the lion’s (אַרְיֵ֑ה) mouth And from the horns of the wild oxen! (v21 NKJV)
As אַרְיֵ֑ה (lion) is used in both verses 13 and 21, it is unlikely that writer's use of a different word (כָּ֝אֲרִ֗י) is intended to convey the meaning of "lion,: and translations which render all three as “lion” obscure a difference present in the Hebrew text.
The Septuagint translates the Hebrew into Greek before the Christian and provides a translation unaffected by a Christian interpretation. As The NonTheologian notes in their answer, the LXX supports “pierced” not “lion;” it also correctly distinguishes כָּ֝אֲרִ֗י of verse 16 and אַרְיֵ֑ה of 13 and 21:
16: ὤρυξαν
13: λέων
21: λέοντος
The LXX scholar(s) who translated the original Hebrew into Greek recognized the difference in verse 16 and they rejected “lion” as the meaning. However, the word in the LXX does not mean “pierced.” It means dig or excavate: [3736 - orussó] So while "lion" may be rejected, it is reasonable to question, as noted in another answer: if the intended meaning of the Hebrew is pierce, why doesn’t the text more clearly convey that meaning?
The OT answer to that objection begins in the New Testament which indicates there were potentially three different types of wounds that could occur at the location of the crucifixion. One type, broken bones, did not occur; two types did happen.
And He, bearing His cross, went out to a place called the Place of a Skull, which is called in Hebrew, Golgotha, where they crucified Him… (John 19-17-18 NKJV)
Crucifixion would inflict the same type of wound to both hands and both feet.1 The second type of wound which occurred was made by a different means and at a different place on the body:
For these things were done that the Scripture should be fulfilled, “Not one of His bones shall be broken.” And again another Scripture says, “They shall look on Him whom they pierced (ἐξεκέντησαν).” (John 19:36-37 NKJV)
The NT writer describes the wound on His side differently (ἐξεκέντησαν) from how the LXX translator described the wounds to the hands and feet (ὤρυξαν). The Scripture referenced in the Gospel is Zechariah 12:10:
“And I will pour on the house of David and on the inhabitants of Jerusalem the Spirit of grace and supplication; then they will look on Me whom they pierced (דָּקָ֑רוּ). Yes, they will mourn for Him as one mourns for his only son, and grieve for Him as one grieves for a firstborn. (NKJV)
In addition to using a different word from Psalm 22:16, the NT writer’s use of ἐξεκέντησαν deviates from the LXX translation of Zechariah and the NT more accurately reflects the original Hebrew of Zechariah. 2
Comparing the two types of wounds which occurred:
Psalm 22:16 Zechariah 12:10
אַרְיֵ֑ה/ὤρυξαν (LXX) דָּקָ֑רוּ/ἐξεκέντησαν (NT)
Hands and feet Side
Nails Spear
Made when alive Made after death
Crowd surrounded Crowd mourned
The significance of the Hebrew in the Psalm is that it describes a different type of wound and it preserves all of the differences between the prophetic statements of the Psalm and Zechariah. It is also noteworthy that both the state of the wounded (alive/dead) and the response of others to the wounded is different in the Psalm and Zechariah and these differences are also preserved in the Gospel. Jesus was reviled before He died and mourned for after His death.
In particular, the use of a word which means dug in the Psalm for those wounds to both the hands and the feet is extremely accurate in describing crucifixion. Not only do the hands and feet receive the same type of wound, the instrument which makes the wound remains in the body; unlike that in the side where the instrument which causes the wound is removed.
Conclusion
First, lion is contrary to how the LXX scholars understood Psalm 22:16. Second, the use of a word which means dug accurately describes the nature of wounds inflicted by being nailed to a cross where the nails also remain in the body. In addition using a word which means dig or burrowed, accurately describes how the wounds would be inflicted on the hands as the person nailing would need to feel the body to position the nail between the bones (not breaking any) and not puncturing a vein or artery.
Finally, the Psalm locates the wounds in the hands and not the wrists (where they most likely were made). However, there is no OT use of the word wrist. The failure to distinguish between hands and wrist is consistent with how the word is found throughout the OT and cannot be a basis for the denying the accuracy of describing the wounds made by crucifying Jesus. While it would likely be criticized as blatant Christian eisegesis, the meaning of 22:16 is probably better conveyed as punctured or driven through rather than pierced. Obviously that reflects the Christian perspective that Psalm 22 is a prophetic statement of the crucifixion of Jesus. Nevertheless that is consistent with Greek translation in the LXX and the actual events.
The intended image of Psalm 22:16 is that of Jesus Christ nailed to the cross, alive, surrounded by others who despise, ridicule, shake their head, saying “he trusted in the LORD, let Him rescue him" and gaping at Him with their mouths; with His bones out of joint; strength dried up; His tongue clinging to His jaws; others staring at Him and dividing His clothes.
1. In addition to using a different word (from vv 13,21), there is no logical reason for including both hands and feet if the meaning is “lions.” This is especially true given the context which describes dogs and companions encircling me.
2. http://biblehub.com/commentaries/john/19-37.htm