There is obviously a lot written on Psalm 23, and this verse in particular, so it is not probable that we will completely solve this issue here.
There are two translational (how do we move it into English) and interpretational (how do we understand the concept of this word) questions:
how do you translate and interpret "restore" or "refresh" which comes from the Hebrew verb שוב
how do you translate and interpret "soul" - Hebrew נפש
The Theological Dictionary of the Old Testament (TDOT) has 61 pages on שוב alone.
The verb form in this verse is Polel. Timothy Willis notes (Willis, Timothy M. 1987. “A Fresh Look at Psalm 23:3a.” Vetus Testamentum 37 (1): 104–6.)
the Polel indicates a plurality of action.
and he translates this verse
"He gathers me in" or "He herds me in"
Willis notes the similar use of this verb (Polel) in Jeremiah 50:19 where Israel is described as a flock that has been scattered and God is going to gather them back in.
I will restore Israel to his pasture, and he shall feed on Carmel and in Bashan, and his desire shall be satisfied on the hills of Ephraim and in Gilead. Jer. 50:19 (ESV)
This translation for Psalm 23 this would describe the three main responsibilities of a shepherd: food, water, shelter. Willis writes,
this interpretation provides a different understanding of the imagery in Ps. xxiii 3a. The psalmist is not describing the physical or (implied) spiritual result of the nourishing which Yahweh provides for his people…Instead,…tells three things which a shepherd commonly does for his sheep (106).
(obviously, this is open to debate since scholars have debated how to translate this word)
The second difficulty is how we translate and understand the Hebrew נפש for "soul." Our modern understanding is that "soul" is the immaterial part of us that departs the body upon death - hopefully, destined for heaven.
The Hebrew concept - again, according to the the Theological Dictionary of the Old Testament (TDOT) - for "soul" is much more in line with your "whole person." You are a soul rather than you have a soul. This is why some Bible versions translate נפש as "life" or "strength." Consider the Contemporary English Version (CEV):
and you refresh my life. You are true to your name, and you lead me along the right paths.
נפש has a broad range of meanings and good scholars on all sides of debates don't always agree.
One this for sure - this is an amazing Psalm which has impacted millions over the centuries. However we read it, God's word is bigger than all of us and He can renew, restore, replenish, refresh, or even return our "soul" "strength" or "life."