Something that bypasses the Line could work.
The Goblet itself doesn’t detect age, which is why the Age Line was even necessary.
“To ensure that no underage student yields to temptation,’ said Dumbledore, ‘I will be drawing an Age Line around the Goblet of Fire once it has been placed in the Entrance Hall. Nobody under the age of seventeen will be able to cross this line.”
- Harry Potter and the Goblet of Fire, Chapter 16 (The Goblet of Fire)
The piece of paper with the underage student’s name on it would just have to get across the Age Line to be successfully entered. However, it’s unclear how exactly the Age Line works, other than wizards who are under the required age can’t pass. Whether it’s also a magical barrier that would stop inanimate objects, creatures like owls, or house-elves could pass over or through isn’t mentioned.
Getting an older student to put it in would have worked.
When Harry’s name comes out of the Goblet of Fire, Dumbledore asks him if he had an older student put it in for him. Since he’d surely know how the Age Line he cast worked, he’d also know the possible ways to get around it.
“Did you ask an older student to put it into the Goblet of Fire for you?’ said Professor Dumbledore, ignoring Snape.”
- Harry Potter and the Goblet of Fire, Chapter 17 (The Four Champions)
McGonagall also seems to consider this one of the only plausible ways Harry could have put his name in the Goblet.
“Really, what nonsense! Harry could not have crossed the line himself, and as Professor Dumbledore believes that he did not persuade an older student to do it for him, I’m sure that should be good enough for everybody else!”
- Harry Potter and the Goblet of Fire, Chapter 17 (The Four Champions)
This is logical - the Age Line stops underage students from crossing, but when someone of age crosses, it doesn’t check the paper to see if it’s actually their name.
Therefore, it might perhaps work if the underage student who wants to compete, if they can’t find a willing older student, could try doing a Switching Spell on a piece of paper held by an older student to replace it with one with their name instead. They might also be able to Confund an older student into putting it in - the Age Line might not detect the mental status of the person crossing. A very devious and talented student could possibly Imperius an older student to do it for them.