I'm not necessarily talking about St. Jude's hospitals, but they're pretty cool too. St. Jude is the patron saint of hopeless cases, lost causes, and hope. He's an awesome guy! St. Jude was one of the original twelve apostles, which means he was one of the first people to preach the gospel and he was a close personal friend to Jesus. St. Jude is technically St. Jude Thaddeus, brother of St. James the lesser, and likely cousin of Jesus. St. Jude went through Mesopotamia, Libya, and Persia preaching the good word with St. Simon. St. Jude died a martyr.
Devotion to St. Jude picked up a lot during the great depression and World War II, which was definitely a time of great hopelessness. Those people, as we do today and as some have throughout church history, turned to the intercession of St. Jude to pray with them and for them as they felt helplessness and desperation. There is never a wrong time to seek his aid. To demonstrate this, I will provide examples from my own life.
Growing up, my step-grandfather was very sick. He had kidney failure and had to receive dialysis several times a week before it was all over. He suffered greatly, and in his suffering, he became closer to God. It was a frequent occurrence to go up to St. Louis from Marshall (where I'm from, small Missouri town) to see Papa Weldon in the hospital because, though I didn't necessarily realize it at the time, it would probably be the last time I saw him every time I saw him. And every time I saw him, he'd give us all St. Jude prayer cards and tell us to pray for him and for us to ask St. Jude to pray for him. His wife, my grandma, was one of the most influential people to the development of my faith. She would read the Bible with me when I was very little and we would pray for Papa Weldon together, and ask St. Jude to please pray for him too. There were so many times that he went into the hospital and came back out despite all probability. Eventually, his suffering ended and he joined God and St. Jude. At his funeral, St. Jude prayer cards were handed out, and nothing could have been more appropriate.
My mother is not Catholic (though she goes to mass every week, often by herself since I've gone to college), but she still greatly encourages devotion to St. Jude. Every time I talk to her about something I'm really struggling with (like a biology test most recently) she'll say "I'll talk to St. Jude, you should too". So I'll pull out my St. Jude prayer card from my grandpa's funeral and ask my confirmation saint, my family's patron saint, and my good friend, St. Jude to talk to the big guy and help me out. I encourage you to do the same.