OK so Houston is obviously a big place but if you get driven to the downtown, it's basically a huge complex of gleaming office buildings and office parks. Yeah, it's kind of the epitome of Texas in bourgie form.
You can totally crash at the Doubletree hotel downtown for free. Go around it, there is a park in the back where there are these gnomes buried increasingly deeply. (don't sweat if you don't see the gnomes) Anyways in the back of the Doubletree, where the park is, there is a concrete staircase. Go up the staircase, that door is often unlocked- its where they have conferences and weddings, and you can just walk in and sleep in an unused conference room, or, if you wanna make sure you don't get kicked out, you can sleep in the stairwell where they never go. A conferance room is nicer but they may be just about to use that room and you'll be discovered. That's what happened to us but we woulda been fine if we had just found a better conference room or slept in the stairway, we kinda were hanging out at the wedding and they started to get suspicious. (Has someone tried this recently?)
Sleeping under the highway is typically possible in downtown Houston. Follow the bicycle trail along the river and choose any spot you like. It is far enough from the highway to sleep in peace and lacks pedestrian traffic.
Houston has plenty of truck stops ranging from $7 to $12 per shower. However, your cheapest bet will be in Memorial Park. They have a running center (whie building just off Memorial Drive) where showers cost $1. Locker and clean towels cost an extra $1.25.
Best bet is to take the commuter bus to the park & ride out in Katy (near highway 99), cost is $4.50. Get a Q card for 5 bucks at any grocery store and enjoy unlimited bus transfers, you'll have to take 3 busses to get there from downtown. It's a couple miles walk around the two I-10 bends, but once you get there it's an easy hitch directly from the freeway. Plenty of good camping in the area, definitely check out Wildcatters.
Commuter bus to Baytown Park & Ride ($4.50) will drop you off at the San Juacinto mall close to the 10. Get a Q card for 5 bucks at any grocery store and enjoy unlimited bus transfers, you'll have to take 3 busses to get there from downtown. Good camping in the area (avoid the mall - security guard). There is a church on the eastbound frontage road with a covered area in case it rains. Near the same church is a decent ramp to hitch from.