Can you tell me what the name of this bike is?

It looks quite similar to the Schwinn Stingray. If you Google around they seem to have made quite a few models, and were quite popular, and hence, many knockoffs were made in order to mooch off their popularity. Your's looks similar to the one in the 1977 catalog, except your's has the gear shifter, and they do list 3 and 5 speed models.

My grandmother worked for Sears for 40 years and almost everything I ever got from her came from there. She got this bike for me for Christmas 1970. It was the sickest bike on my street and was stolen by the end of the year.

I have been looking and believe it might be a Sears Spyder 500 5-speed?

It has to be a Huffy Rail. The decal on the chain guard is exactly the same. All the geometry, too.

That's definitely a Huffy/Sears "The Rail" 5 speed. My got one in '65 in 6th Grade. In Purple. Not an exactly un-cool color, but border line. I was doing a Google search for my old bike and came across your post from 6 years ago and couldn't stop myself from adding 2 cents worth. That tall bar behind the seat was called a "Sissy Bar" back in the day and a necessary mod if you wanted to be cool, lol. Hope I didn't bore you.