As that is one way to interpret such scripture, I thought it would be interesting to take a different approach to the concept, by going to its origins in the Old Testament, and see what is meant by the phrase.
The earliest verse that suggests this concept is in Isaiah 5:26-27
And he will lift up an ensign to the nations from far, and will hiss unto them from the end of the earth: and, behold, they shall come with speed swiftly: None shall be weary nor stumble among them; none shall slumber nor sleep; neither shall the girdle of their loins be loosed, nor the latchet of their shoes be broken
Here we see in First Isaiah that the concept of not being weary has to do with coming swiftly to the Lord. In this instance, an ensign is lifted to the nations, to which the people hasten to unwearingly. For Latter-day Saints, the ensign to the nations is the fulness of the gospel of Christ, found in the Restored Church. While we see many joining the Church now (more than 250,000 per year), the day of reckoning shall approach, where millions will joyfully flock to the banner of Zion for safety and refuge (D&C 45:66-71).
For Mormons, it is akin to those who flocked to Captain Moroni's banner of liberty, in order to defend their homes, family, religion and freedoms (Alma 46). Moroni created the standard of liberty from his cloak, and marched throughout the land calling others to join him. Many rushed forward and tossed their own cloaks at his feet, as a symbol that they were willing to give all to establish and protect that ensign to the Nephite nation.
Jeremiah shares this thought with us:
But thou, O Lord, knowest me: thou hast seen me, and tried mine heart toward thee: pull them out like sheep for the slaughter, and prepare them for the day of slaughter.
How long shall the land mourn, and the herbs of every field wither, for the wickedness of them that dwell therein? the beasts are consumed, and the birds; because they said, He shall not see our last end.
If thou hast run with the footmen, and they have wearied thee, then how canst thou contend with horses? and if in the land of peace, wherein thou trustedst, they wearied thee, then how wilt thou do in the swelling of Jordan?
For even thy brethren, and the house of thy father, even they have dealt treacherously with thee; yea, they have called a multitude after thee: believe them not, though they speak fair words unto thee. (Jeremiah 12:3-6)
Jeremiah could be speaking to the Lord or his fellow Israelites in stating that if they are wearied by their own people in Jerusalem, how can they be happy elsewhere (and in particular, Zion)? If they are not happy when living in a peaceful land, then how will they be happy when the floods (from the Jordan River) come and ravage the peace and prosperity they do have?
In this instance, they are not called to seek after Zion, because they are already supposed to be there! Yet, in establishing David's kingdom, the temple, and what was to be a holy people, Judah was not Zion. It still leaned on royal kings, other nations and gods, rather than having Yahweh as their Lord and King. They were not happy with the best God had to offer them, and so they lost everything. All the markers of a Zion people: temple, priesthood, the covenant (ark of God's Presence), were lost or diminished because the people grew weary of the things of God.
The most common and most easily spotted version of the phrase is in Isaiah 40:31:
But they that wait upon the Lord shall renew their strength; they shall mount up with wings as eagles; they shall run, and not be weary; and they shall walk, and not faint.
In this first chapter of Deutero-Isaiah (probably written by one of Isaiah's followers), we see that it definitely sounds like a strengthening of physical health? But let's look closer at the .context.
Hast thou not known? hast thou not heard, that the everlasting God, the Lord, the Creator of the ends of the earth, fainteth not, neither is weary? there is no searching of his understanding.
He giveth power to the faint; and to them that have no might he increaseth strength.
Even the youths shall faint and be weary, and the young men shall utterly fall:
But they that wait upon the Lord shall renew their strength; they shall mount up with wings as eagles; they shall run, and not be weary; and they shall walk, and not faint. (Isaiah 40:28-31)
Those who actively seek after God do not just receive good health, but they receive an attribute of God: they do not grow weary and faint! God, who has created the earth, does not tire. We cannot find out all that he knows by searching, suggestive of the Word of Wisdom's promise of secret treasures of wisdom and knowledge.
In the world, the time will come when the hearts of men will fail. Even the young, who are normally full of hope and energy, will become weary because they have not sought eagerly after the Lord.
But those who run or walk eagerly to join the Lord shall receive not just good health, but HIS strength. As he is not weary, they will not be weary, either. God will impart his strength. They will be lifted up on wings of eagles, suggesting being carried to a celestial level of holiness. It suggests action or movement toward God.
In D&C 84:33-34, it notes that when we accept the priesthood and magnify our callings (something both men and women can do), we become the children of Moses and Aaron, and the seed of Abraham. In rising up to a new holy lineage, we are then "sanctified by the Spirit unto the renewing of their bodies" and become "the elect of God."
While we often see the renewing of our bodies as giving us good health, so we can live long lives and serve, may I suggest a better interpretation: we are changed from telestial beings to a higher spiritual and temporal sphere of existence. Our spiritual body, connected to a physical body, will be changed from the natural man to the spiritual man, even the man in Christ. In so doing, we become as God: not weary or fainting, but able to overcome all things and do great things, even as the Creator does such marvelous things as well.