I live across the street from JSC, and have for 20 years.
The following will make more sense if you print out a map that shows the various towns, and streets.
You didn't say if you were actually going to work on site at JSC, or for a NASA contractor. If you're working on site, all the entrances are along Nasa 1 (main East/West street), to the south of the site. I think the site is only running 25% capacity now - they'll stretch it out for as long as humanly possible to avoid having to go back to work in the office, as the NASA workers like working from home.
You didn't mention kids or schools, which will play a huge role in where you live, as different districts have different quality schools. I'll assume schools are not a factor in your house location.
Another factor will be do you want water view / access? NASA is next to Clear Lake, which connects to Galveston Bay and then the Gulf. Water view places are obviously more expensive, and much fewer to be had. I don't live on the water but across the the street from it, have water views to the sides, and it is nice seeing the water and all the pelicans / gulls flying around.
People have mentioned Pearland and Friendswood - those are two nice suburb towns to the west - but it's a good 20 minute drive, no traffic, from JSC to Friendswood, and probably 30 to Pearland. And Nasa 1 back up badly in the evening going west. I'd wager if you left at 5 it'd be 30-40 minutes to Friendswood.
Across Nasa 1 is Nassau Bay, another nice suburb town. Like the other towns around the Clear Lake Area, it has its own city government, police, mayor, city council, so you don't have to deal with the Houston jokers, and HPD which won't have the response time, or care about factor that a small town outfit will for things like noise violations, traffic accidents, car burglars, etc. It's a mix of older houses, ones that have been updated, and bigger ones here and there, and if they have water views to warrent a larger house.
Down the road to the east are various smaller towns - Taylor Lake Village has some nice houses in it, then Seabrook goes to Galveston Bay. Kemah is over the bridge, and to the South, on the other side of Clear Lake. Traffic for Kemah is going to cause a longer commute than the other places.
League City is also south of Nasa, and again, it's a longer roll than Nassau Bay / Taylor Lake / Seabrook. Webster is a bit further West of NASA, but on the same side of I45. It's an older neighborhood and generally not as modern and large houses as the other places.
The nicest places around are in or around Bay Oaks Country Club - it's to the Northeast of NASA. Some really nice houses along the greens, and generally pretty upscale. Those places I believe are in City of Houston, as is NASA itself, though on a long term lease to the Federal government is my understanding. I've not lived in the City so I can't comment on their level of services like police, water, roads, etc. You don't really notice a difference just driving around. But you'd think a local government would be more responsive than a huge one like Houston. Then again, maybe smaller towns have money issues for fixing stuff - can't tell unless you know all the details about place X and place Y.
Clear Lake refers to the lake (actually it's connected to Galveston Bay and isn't a lake), and the general area of East of I45, south of Beltway 8. There's not a Clear Lake city government, as it's all City of Houston, though I think they have their own water board or such.
One guy mentioned Hurricane Harvey and flooding. Harvey dumped 50-60 inches of water over a few days here. I'd never seen it rain like that - the sky just flowed water like a waterfall. Many places that had never flooded before, flooded then, so I would not auto disqualify a place that flooded during that storm - 50 inches of rain in 2 days is not a usual occurrence. But you do want to know, both as a guide for possible other storms, and for finding out if the place was properly repaired if it flooded.
Another thing to find out is elevation above sea level, and if the place floods or gets close during hurricanes. Hurricane Ike in 2008 put a foot of water in my garage, and was 2 feet from getting up to my living room. Living room is at 12 feet above sea level as a guide.
Places close to the water are going to have high insurance premiums, so ask what their current policy costs and covers.
There's an awesome private range in Friendswood, called Pearland Shooting Center. There's a waiting list to join, but if you move definitely sign up and try to join. 200 bucks a year or so, all you want to shoot, many many rifle and pistol ranges.
Clear Lake area is about 20-25 miles to downtown Houston. Traffic is bad during rush hour and best avoided, but at other times like the weekend, it's maybe 30-35 minutes there. Hobby airport is about 15 miles away, and between Southwest who had a big hub there, and Delta to Atlanta, you can go most places via that one, and not have to go up to Intercontinental, north of Houston and a 45 mile drive.
It's a very nice area - no real slums or public housing. Webster has had a lot of illegals move in the last 20 years for work at restaurants and the mall and such, as well as lots of construction jobs, but they don't seem to cause any problems, at least to us home owners - don't know what it's like to live there in one of the apartments.
Houston has a very high level of restaurant quality, and there are many great places to eat. Weather is hot May to September time frame, but nice the rest of the year.
Gun stuff will be a breath of fresh air compared to WA - open and campus carry passed a few years ago, and we're on the cusp of getting Constitutional Carry passed. The gun grabbers are playing defense here instead of offense like it WA.
I think you'll like it, and would recommend you move.
Holler at me with any other questions. Good luck.