The absolute closest airport to MD Anderson/Texas Med Center would be Houston Hobby airport (HOU)... the down side to this is that it's a pretty busy airport. It's in Class B airspace with a good bit of airline and corporate traffic; not really a big deal if you're comfortable yakkin' on the radio. If you'd prefer to avoid all that hubbub and the high FBO prices, the next closest would a toss-up between Sugar Land (SGR), Houston Southwest (AXH) or West Houston (IWS).
West Houston is uncontrolled, easy in and out, has quite a bit of flight school traffic, FBO parking on the W side of the field. Highway access to the Med Center might be a bear, I-10 to the 610 Loop to 59 N... ugh.
Sugar Land is controlled but pretty quiet and laid back, also easy in/out. Big runway with a brand new terminal and FBO facility. Highway access to the Med Center would be fairly easy; Hwy 59N.
Houston Southwest is a decent little airport, but I don't get in there much due to the coarse runway pavement. It just seems a bit harsh on my tailwheel... noisy! Watch for the antenna farm a few miles N of the field; they top out at 2100' msl. A north arrival here could be a bit of a handful on a hazy day while staying below Class B and threading the needle through the SGR class D airspace and dodging those antennas. It's not a big deal, just keep your eyes open.
Hope this helps...