When I was a kid and we were homeless for a short time, we lived in van at a KOA right off of Beltway 8 and Aldine Westfield Rd, which is about 5 miles from the Bush Airport (north side of town).
Not sure if it is still there, but if it is, it would be a good and cheap location. If long-term is a week or three.
If you want longer term, google search RV storage in the towns of Humble, Spring, Atascocita. There are tons of them. Some are enclosed, some are covered, some are just out in the open with other people's boats and RVs.
Any of these towns would have shuttles or taxis to get you to the Bush Airport.
2015 24' Rockwood UltraLite 2304DS behind '14 Diesel Jeep Grand Cherokee/'04 Hemi Ram 1500. Equal-I-Zer 4-Point 1000lb sway control.