I used to live in Henderson, next door to Vegas. Even though I didn't venture downtown to the strip too often, I've been to many of the hotels/casinos.
Excalibur has some good kid stuff, fun things to look at, a dinner that your little one will fit right in at (you eat with your hands LOL!).
Circus Circus has all the games, a midway, baby rides in the fun park (in the pink dome) and a circus act. I DO NOT recommend the buffet at Circus Circus, it's GROSS! However MGM has an awesome buffet as does the Rio (Rio is a bit off the strip). I'm not certain what New York New York has to offer as I moved out of there before I had time to get to know it.
It's still not warm enough for the swimming pool, but I believe a couple of the hotels may have indoor pools, you'll have to call around on that.
Of course, your little one will enjoy the stroller rides the most, since walking is the transportation of choice out there

However, you are not allowed in the casinos at all with kids under 21, so don't even consider walking with him AND planning on gambling at the same time. He cannot even sit in the stroller when you sit at a slot machine, they will let you walk through the casino, but don't stop! They will say something to!
Have fun, but keep in mind your little guy may be too small to really enjoy any of the attractions, so you may want to just stick with the buffet or McDonalds and let Grandma and Grandpa enjoy his company