1. They automatically walk well on leash.
Greyhounds have such a high centre of gravity that they simply can't pull on the lead like a dog which is lower to the ground. I have had Jack Russel Terriers who can pull harder on the leash than a grey.
2. They don't smell like dog & they are easy on allergies
They don't produce much body oil, and as such don't have that 'doggy' smell that most other breeds have. As they are not as oily and don't shed as much dander as some other breeds they can be good for allergy sufferers. The disadvantage to this is that they can tend to have dry flaky skin if they do not get enough oil in their diet.
3. They are awe inspiring at top speed
A greyhound can reach speeds of 45 kmh in 3 strides and full speed of 70 kmh within 30 metres or six strides. The only animal that accelerates faster is a cheetah. The downside is that you can't let them off to run anywhere near traffic.
4. They are very sensitive and intelligent dogs
Sure, you can't expect them to do the same trick over and over, because they will get bored. You can't train them the same way as you can train other breeds, but you can still train them quickly and well. They need a soft hand. The only thing that will be unreliable is recall, particularly in new places where there are prey like animals and other distractions. They have been used as therapy dogs for soldiers with PTSD because they are perceptive and gentle and can identify when PTSD sufferers are dissociating.
5. They adapt to your activity level
Greyhounds are not 'stayers'. Their 'long' races last about 45 seconds, and this is what they are bred to do. They are more than happy to sit on the couch all day, and have a quick walk at night. But they also adapt to longer more regular walks, if that is what you want to do with your dog. Especially if you get a young one with plenty of 'puppy spunk'.
6. They are gentle
A greyhound is built for speed but not for impact. It is pretty unlikely that your greyhound is going to knock you over like a labrador or golden retriever will at full speed - after a few collisions they learn to avoid them. Even though greyhounds are big dogs, it means that they are unlikely to knock a small child over like I have seen many other dogs do.
7. The do not bark when they are bored
Greyhounds don't seem to get bored easily. Maybe it's because when you aren't interacting with them they are happy to be asleep. Some greyhounds will bark if they want to be let inside, while my girl has learned to bark if someone comes to the door (but not if there is someone she knows in the common driveway). She only does this because my other dog does, but he is a bit of a hair trigger, so I don't bother getting up unless she barks as well.
8. They are very passive and easy to handle
Vets & vet nurses love them. It is a good thing too because they don't deal well with general anaesthetic because of their low bodyfat levels. The vet restitched Barbie and then took out her stitches from her sterilisation procedure without her even batting an eyelid. There was no desensitisation training required. Vet nurses often tell of their greyhound blood donors who accept a catheter in their artery without even squirming!
9. They are dog-social and trustworthy on outings with many dogs
Aside from the rare damaged individual who is a little freaked out by non-greyhound canines, greyhounds in general are very dog-literate dogs. They are bought up with their littermates and spend a lot of time in their company. My girl has taken a couple of months to learn that not everyone wants to play chasey at full greyhound speed but she is very good interacting with others.
10. They are loving and affectionate in a non-pushy way
She won't force her way onto your lap (though some greys might rest their head there) or force her head under your hand when you are trying to type, but she will always sit next to you if there is sufficient space and keep an eye on you around the house without getting under your feet.