If mere luck is not responsible for Bilbo's success, what is?
Although luck is a factor, most of Bilbo's success can be attributed to how clever and capable he is. His whole journey is started because Gandalf believes that he is the most qualified person (Hobbit) for the job; and even though he does not believe it at first, he eventually lives up to Gandalf's expectations because of his cleverness.
The first really clever thing he does is when he and the dwarves are captured by the goblins. He gets separated from his companions and ends up in Gollum's cave. This is an example where luck plays a part (ie. Bilbo accidentally stumbles upon the One Ring), but it is his cleverness that saves him from Gollum and gets him out of the goblin tunnels (ie. the riddles in the dark and tricking Gollum into leading him to the exit).
His next really clever moment is in Mirkwood, when he saves the dwarves from the spiders. His luck is that he has the One Ring, his cleverness is that he uses it to hide while he sings songs insulting the spiders, throwing rocks and using Sting to kill them. Also in Mirkwood, Bilbo uses the One Ring to sneak around in secret, eventually freeing his companions from their prisons.
Another of his most clever moments is with Smaug. He had learned after Gollum not to give out his name or any other information like that, so when Smaug was asking him questions in an attempt to figure out what Bilbo was, Bilbo only answered in riddles. And when Smaug left the mountain to go destroy Lake Town, Bilbo sent a thrush (a bird) to tell Bard the Bowman of Smaug's weak spot.
In the end, the biggest part luck played in Bilbo's successes was in the form of the One Ring: because he found it, he was able to do the things he did. But simply being invisible was not enough to make his plans work; he also had to be clever and resourceful.