TL;DR There's no significance to it, it's just following tradition of naming a child after famous ancestors.
Tradition of naming Children after ancestors
Although People in different regions of Westeros share some names but on the other hands we have some names which are exclusive to some regions.
House Lannister is from Westerlands. Therefore, they tend to give their children traditional Westerlander name. Also, another factor to consider is that Great Houses often name infants after some ancestor which brings a sense of continuity to the name.
For example, Brandon Stark is named after his uncle Brandon Stark. Brandon Stark (The uncle) must have been himself named after several Stark Kings named Brandons such as Brandon the Builder, Brandon the Shipwright, Brandon the burner, Brandon the Daughterless etc.
Historical Members of House Lannister with similar names
When we look at historical members of House Lannister, we see:
- King Tyrion I of the Rock
- King Tyrion II of the Rock
- King Tyrion III of the Rock
If we look simply for historical Lannister with names starting with the prefix Ty, we see:
- King Tybolt Lannister
- King Tywell I Lannister
- King Tywell II Lannister
If see more recent members of House Lannister, we see:
- Tytos Lannister (Note there is a Riverlander Lord, Tytos Blackwood with the same name)
- Tywin Lannister
- Tygett Lannister
- Tyrek Lannister
- Tyland Lannister
So as evident, Tyrion's named after great men of House Lannister just like many other members of other great houses are named after their ancestors. The prefix Ty seems to very common in Lannisters and appears to be unique to Westerlands.
Who named Tyrion and Why?
It is not clear who named Tyrion. His mother before dying? His father afterwards? Or someone else? So given that we do not know who named him, we can't speculate the motives behind this act.
I have been doing bit of reading the Citadel transcripts of GRRM's correspondence with the fans. I stumbled on something GRRM said about who named Tyrion.
It was most likely his father, Tywin. From So Spake Martin, Correspondence with fans:
Q: Since all of their mothers died, who gave Jon Snow, Daenerys Targaryen and Tyrion Lannister their names?
George: Mothers can name a child before birth, or during, or after, even while they are dying. Dany was most like named by her
mother, Tyrion by his father, Jon by Ned.
Apparently Tywin doesn't hate Tyrion?
Tyrion's father doesn't like Tyrion because:
- Tyrion's birth killed love of Tywin's life.
- Tyrion's malformed and thus doesn't fit Tywin's high standards for Lannisters.
- Tyrion's deformities made Tywin a target of general ridicule and laughter. Tywin absolutely hated being laughed at.
- Tyrion married a commonborn girl named Tysha, thus making Tywin further ashamed.
- Tyrion is known to be excessive in his carnal tendencies and frequents whores. Tywin hates that as well.
- Tywin suspects Tyrion to be someone else's child so there's that as well.
So it is unlikely that even if Tywin named Tyrion, he had some special affection for the child.
Jaime and Cersei
As for Jaime, the only other character in Planetos that I can recall with the same name is Jaime Frey who is of course a Riverlander unlike our beloved Kingslayer. For Cersei, the only other character with the same name is Cersei Frey. These kids also have a brother named Tywin. It appear that Ser Raymund Frey named his kids to win favor of the ruling Powers.
This doesn't mean anything however. Tywin himself had following siblings:
- Tygett Lannister
- Gerion Lannister
- Kevan Lannister
- Genna Lannister
As you can see, only Tygett shares the "Ty" with Tywin while all other siblings have different names. Not to mention, Tywin's father Tytos' father's name was Gerold. There have been three Lannister Kings named Gerold and so Gerold was named after those men.
If all Lannisters were "Ty" for more than 12,000 years of their history, I suspect it would have caused much lulz for rest of the Westeros along with a lot of confusion.
Anything from the books?
Although there's no explicit significance to Tyrion's naming or anything else in the books, there's one Quote from Genna Lannister which you might find interesting in terms of Tyrion's uniqueness as compared to his siblings:
Jaime, sweetling, I have known you since you were a babe at Joanna's
breast. You smile like Gerion and fight like Tyg and there's some of
Kevan in you, else you would not wear that cloak...but Tyrion is
Tywin's son, not you. I said so once to your father's face, and he
would not speak to me for half a year.