Offering it Up
So, how do Catholics "offer up" their sufferings? In both formal and informal ways.
Formally, many Catholics make the Morning Offering to give to Our Lord that day's efforts, works, joys, sufferings, intentions, etc. (the form may vary). At the Mass, we excercise our lay priesthood by consciously, silently, privately offering ourselves up, along with the Son, to the Father during the Offertory.
Informally, we "offer it up" by simply asking God in our own words to use a suffering as it occurs. We often do this for specific intentions (ex., "Use this pain, Lord, for the salvation of my sister/brother...").
It's quite a discipline to react to suffering this way! In mental or physical pain? Drop something on your toe? Putting up with a co-worker who is making your life a living Hell? Enduring the constant ache of arthritis? Standing in line at the grocery and hating every minute of it? Spill the milk? Accept these things in peace, and ask God to use them for the good of the Church or for a more specific intention close to your heart. This isn't easy to do (and I in no way claim to be good at it), but it does make the suffering more meaningful and less -- well, less insufferable!
You'll find that it is not uncommon to hear one Catholic tell another who is suffering to "offer it up" as a way of dealing with his suffering. It should be remembered, though, that while it is most definitely good to tell someone to "offer it up," it is also easy -- and that we are called, too, to comfort those who are suffering, to feed the hungry, to give drink to the thirsty, care for the sick, etc. Telling someone to offer it up without also helping him to deal with the temporal and emotional effects of whatever they are going through is not the fully Christian thing to do.
And always help the suffering to retain (or regain) Hope that his suffering is not in vain. Assure him that he will partake of "the consolation":
2 Corinthians 1:5-7
For as the sufferings of Christ abound in us: so also by Christ doth our comfort abound. Now whether we be in tribulation, it is for your exhortation and salvation: or whether we be comforted, it is for your consolation: or whether we be exhorted, it is for your exhortation and salvation, which worketh the enduring of the same sufferings which we also suffer. That our hope for you may be steadfast: knowing that as you are partakers of the sufferings, so shall you be also of the consolation.
Another verse for those who suffer:
For the Spirit Himself giveth testimony to our spirit that we are the sons of God. And if sons, heirs also; heirs indeed of God and joint heirs with Christ: yet so, if we suffer with Him, that we may be also glorified with Him. For I reckon that the sufferings of this time are not worthy to be compared with the glory to come that shall be revealed in us.