What is he anxious about? Is it the drop or speed of coasters, or the fear of motion sickness or the jerking action of some rides.
I think this is a good question. What is it he doesn't like about rides? For instance, I will ride BTMR, Splash Mountain, EE, or Manta (Sea World) all day, but don't really care for RocknRoller Coaster or Space Mountain because they seem to jerk me around more. I also don't like to ride if I can't be up near the front because the jerking seems to be worse in the back.
Most of Disney is very tame from a roller coaster standpoint.
You've really only got 5 true roller coasters to worry about:
Expedition Everest (AK)
- intense, fast - 50mph, starts out ok, but then suddenly, goes backwards very fast, gets very high off the ground, is 200 ft tall, has an 80ft drop, outdoor coaster which goes through some dark cave sections. It freaked out my friend whose a guy and says Barnstormer is more his speed.
Rock n Roller Coaster (HS)
- intense, fast - 57mph, 5G's, only Disney coaster to go upside down (twice, plus one corkscrew), indoor coaster, black lights in a dark room, can't tell how high you are due to lighting.
Space Mountain (MK)
- intense, seems faster than it is due to the intense darkness only 30mph, black lights in dark room, cant really see track, doesn't go upside down but do to extreme quick turns and the pitch blackness you may feel as if you are, it's very disorienting and hard to tell which way you are going, one 35 ft drop.
Big Thunder Mountain Railroad (MK)
- less intense, outdoor coaster, doesn't get too high off the ground, although a few sections you are up in the air, 45ft height, consistently goes about the same speed 30mph, has a few intense drops and turns, but overall much tamer than the first three mentioned, still pretty fast though, not a kiddy coaster
Barnstormer
- less intense than the railroad, outdoor coaster, not too high 30ft, not extremely fast 20mph, short ride less than a minute with only about 20 seconds of that being the ride part of the coaster, as a-posed to loading/unloading, positioning, mainly small turns and dips, more of a Jr. coaster. I wouldn't call it a kiddy coaster because to me those are the really small ones that go in a little circle with just a few humps. This is a coaster, just a much smaller very short one.
Beyond the roller coasters the following are other rides that are seen by some as intense, scary, or that cause motion sickness
Dinosaur (AK)
- bumpy, jerking, dark room ride, where things pop out at you. There are unexpected drops and turns, the ride starts out slow than gains speed towards the end. Main issues people have with it are fear of man eating dinosaurs, especially in young kids, and motion sickness due to jerking.
Primeval Whirl (AK)
Typical crazy mouse carnival ride, jerky turns but relatively tame
Tower of Terror (HS)
Free fall indoor ride, but really it isn't a "free fall" since the ride actually falls faster than the speed of gravity, you wear a seatbelt that literally pulls you down as you fall, the ride is 199ft tall, you are sort of sling shot up and down it for about 3 minutes, the ride drops randomly, sometimes you go all the way to the top, then fall, sometimes you only move up a few floors before falling, doors open and close randomly to give you staggering view if the park from above, although everything happens so fast you don't have time to take much of the view in. Turns people off due to the height, speed, and motion sickness. I personally think it jerks you down as you "fall" and is painful to me because of that. Even though I like other similar free fall rides that actually let you fall, I can't stand ToT as it always causes me pain, but I know others who it doesn't hurt at all, so it just may be me.
Star Tours (HS)
Bumpy 3D simulator ride can cause motion sickness but not scary unless Darth Vader freaks you out.
Mission Space (Epcot)
Simulator ride, issues are claustrophobia due to the very small inclosed seats, you are inclosed on pretty much all sides, and motion sickness, there are two versions - green no g-forces, orange with g-forces similar to those experienced by real astronauts leaving earth's atmosphere
Test track (Epcot)
Track car ride, most of it goes slower, but some areas speed up, curve fast, or are extra bumpy or jerky for effect, you are "testing" your car, although the old version made this more obvious, there is only one really fast spot at the end, where your speed reaches 65mph for like half a second. This would probably be a good first "roller coaster" with only the one really fast short part at the end.
Soar'n (Epcot)
Hang gliding simulator ride, very calm, not scary unless you are afraid of simulated heights, but can cause motion sickness
Splash Mountain (MK)
Water log ride, a few small dark drops indoors, but no scary theming, 1 large 5 story drop at the end.
Tea Cups (MK)
Spinning can cause motion sickness
Astro Orbiter (MK)
Very heigh off the ground, can see most of park from the ride as well as some of the nearby hotels, not good for people who fear heights, fast but only goes around in a circle similar to dumbo just faster, so it can cause motion sickness as well.
I'd hit all the safe rides not listed here first.
Pirates of the Caribbean, Toy Story Mania, Kilimanjaro Safaris, Spaceship Earth, to name a few
Then the ones marked above just for motion sickness if that is not an issue for him
For instance, Soar'n, the Tea Cups, Star Tours, and any rides that just go in a circle (Tricera Tops Spin, Magic Carpets, Dumbo)
Then go a little bit more intense try Test Track, Splash Mountain, Dinosaur.
Which should get you ready for Barnstormer and Big Thunder Mountain Railroad.
If those go ok, then try the other real roller coasters based on what will freak him out the least, going backwards, being up high, upside down, in total blackness, etc. Save ToT for this last grouping as well.
And until you know how he will respond to g-forces , I'd stick with Mission Space green.