Well, it's Will's read on the situation that Waymar Royce is there because he's the third son, but you might want to consider that Waymar also seems pretty keen during the Prologue to have his ranging be a 'success', and to 'find' the Wildlings. Contrast that with Will, who is only there because he was caught poaching, or someone like Chett on the Great Ranging, guys who just want to get the fuck out of there, and you get closer to the answer as to why he went willingly.
Waymar is on the Wall because he wants to be on the Wall. He likes the idea of being a Ranger and tracking Wildlings, he's arrogant and he brings all the wrong sorts of equipment (his sword is all wrong for the deep forests, his destrier is all wrong for the Ranging). The idea of gallavanting around the Haunted Forest chasing savages likely appeals to him. It's why he doesn't listen to Gared or any of the other senior Rangers, because he already has an idea about how this job is supposed to go, and he's not going to let anyone stop him.
Because what other outlet would you have for that passion for a soldier's career at this point? At this time in Robert's reign, there's really no internal or external strife. No significant enemy to fight, besides I guess the Wildlings or the Clans of the Vale, and the latter there's no institution you can join dedicated to fighting them.
You could participate in a tourney as well I guess, when one happens, but we see plenty of scorn from the more grizzled sellswords and soldiers in the series about how fighting only in tourneys is usually for pussys.