It's not just the neck but the entire torso that needs to delevop sufficient strength to support its own weight.
The practical measure is that when a child can raise itself to the sitting position by itself, then the upper body (spine, muscles, etc.) is sufficiently developed for upright load. If you sit the child on the floor and he remains sitting - that's not enough, that's cheating, and you may be putting the child under undue stress if you go biking at that point.
Bike seats (which are mounted using a bracket on the bike's frame) offer sufficient suspension (ca. 3 cm / 1 inch) that minor bumps are evened out. Most bike trailers don't have that much suspension, if any at all! Usually, the only shock absorbing factor is the tires, and they don't offer much. So when comparing bike seats to bike trailers, I would err on the side of caution and add a few months' worth of development.
Once the child is old enough to make himself sit comfortably, he will know how to position himself in a trailer. But you should consider whether the trailer offers sufficient support (and crash protection!) for a child that young, as most are geared toward 3-6yo kids.
The child must sit, not lie down - that would be seriously uncomfortable at the slightest bumps. With a trailer, bumps are felt much more clearly than with a bike or bike seat. Roads that are "so-so" on bike are already "no-no" with a bike trailer. Roads must be nice and flat, and no ruts or potholes. If there are, go over them at no more than walking speed.
Since Michael brings it up: Our son (now 20 months) disliked any kind of hats during the winter, but he has accepted the helmet to the point that he won't even touch his trike without wearing the helmet. I would recommend that a child should always wear a helmet when biking even when in a bike seat or a trailer, from age 0 to 12 years. We have made it a role-modelling policy that the whole party must wear helmets, otherwise nobody bikes. We just completed a 5-day, 330 km, bike trip with my mother who always tries to "forget" the seatbelt in the car, but with respect for our son she too wore a helmet the whole time.