Looking for accuracy, cost-efficiency, and speed: this could provide useful information for determining individual bioinformatic calculations, pertaining to, for example cell type estimation, for personalized medicine regimens. I'm not sure why this question was marked as off-topic and closed. The biological implications are grandioso.
How cost efficient does it has to be? Do you want these data to be comparable to another study?
What do you call surface area?
The interior of the guts (interior of the mouth, stomach, intestins, ...) are typically considered as exterior to the body. I would assume you don't want to consider these things in your measurements, right?
By aware of the coastline paradox
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines.
Similarly, for the surface of the human body, the results will be drastically different depending upon the scale of measurement you want to consider. If you want to measure the small bumps of each atom or if you're happy to consider the the skin is a perfectly flat surface, the answer will vary a lot (numerous order of magnitudes)!
So, if you want to make your results comparable to those of another study, then you should definitely copy their methodology. If all that matters is that you can compare the different individual you measure, then you can pick a methodology that works well for you.
Those are in no way an exhaustive list.
3D scanner is probably the way to go for accuracy.
Approximation of body shape into cylinder and spheres
If you want cheap and inaccurate, you can assume each body part is a cylinder. Consider for example reducing the human body to 4 limbs (which you could further split in two separated by the knee and the elbow), a torso and a head. Just measure the perimeter of each body part and and approximate the surface area.
It might sound like a silly technic but it might well be good enough for your purpose (esp. if you're not comparing your results with another study).
Following Walsberg & King (1978) (thanks to @bshane for pointing this paper out), you can wrap people in a polyethylene film and then measure the area of the film. When measuring the area of the polyethylene film, you are going to have to make some approximation into some simplified (or series of simplified) shapes just like I suggested above. Walsberg & King (1978), when measuring birds, assumed the overall shaped was a prolate spheroid with a hemispherical head. You're going to have to try to find what simplified shape ressembles most to the polyethylene film.
Of cours, you might want to eventually consider other fabrics than polyethylene film and you might want to consider different number of fabrics.
You could prepare pre-measured rectangles and triangles of a given fabrics with colors matching a known size. Then you can just stick those fabrics on the individual body and simply add up the size of the shape you add on the body. This sounds like a difficult solution as you will have a hard time to correctly place the fabrics to avoid wrinkles, overlaps and gaps.
Test reliability of your measure
Reliability of your measure
Do not forget to repeat measurement on a few individuals to measure the accuracy of your measure.
If several people are going to work on this project, you should also make sure to test the inter-raters reliability
- wrap my body in non-overlapping pieces of fabric* and adhesive tape;
- unwrap my body and "ouch";
- measure the area of each piece;
- sum these areas.
*Body-fitting tights, long-sleeved T-shirt, mask, and gloves would cover most of my body quite quickly. The problem is only that i would have to destroy the clothes by cutting them into pieces between the steps 2 and 3.