Most translations of this verse use the phrase "I will give you rest" rather than "I will refresh you." A quick (and hardly exhaustive) search suggests that only the Douay-Rheims and Wycliff versions use "refresh." The rest, from the Tyndale and Authorized (aka King James) versions to the New Revised Standard and Common English Bibles, use rest (no pun intended).
While I don't want to get into some kind of slog about how the verse ought to be properly translated, the difference between "refresh" and "rest" seems significant. I've always loved this verse (usually reading it with the "rest" being given), hearing it as an invitation to lay my burdens down at the altar, to give over the heavy stuff to God when I am becoming worn out by those burdens. And there is an implied relief that one receives upon laying down those burdens. Sweet relief.
But refresh means that I am given something in addition to relief. I'm given what I need to go on. Maybe I'm still laying down those burdens and feeling that sweet relief, but receiving refreshment gives me something new, something besides rest and relief. Refreshment means I may gain new insight, new energy, new commitment, and new power to continue my journey, trusting that more refreshment is always available when my soul begins to flag.
My guess is that I may read this differently, depending on my own state of mind and body and soul. There are times when I just want rest, and the thought of going on doesn't seem desirable or even possible.
But going on is what we are usually called to do. And so there is God's refreshment, like manna in the wilderness that is available every day, just for that day, as a gracious gift. No doubt that's the purpose of its placement in the Rite I Eucharistic service, just prior to communion, when we will again receive refreshment, the power to go on, in the sacrament of Christ's Body and Blood.
As we continue moving through Lent, the season of wilderness and abstinence of various kinds, I will keep the idea of refreshment in mind, not only in my own discipline and journey but also so that I will be mindful that when I celebrate the Eucharist, I am offering Christ's refreshment to those who come to the altar rail with hands outstretched to receive it.