Mechanical engineer here. The answer is it depends. The glue strength, wood strength, and beam dimensions will all play a factor in determining which orientation is stronger.
For Eastern White Pine (PDF) , modulus of rupture is 59,000 kPa.
Titebond II lists its bond strength (I'm assuming shear) as 26,500 kPa.
You'll often hear that the wood will fail prior to the glue failing, which is true for stress perpendicular to the grain. Eastern White Pine's cross-grain modulus is a mere 2,100 kPa.
Beam stress is calculated by moment (torque) on a beam section, multiplied by the distance from the neutral axis (center of beam for a rectangle), and divided by the cross sectional moment of area, "I". For a point load at the center of a simply supported beam (person of weight "W" in the middle of a balance beam of length "L"), the maximum moment will be 1/4*W*L.
The moment of area is equal to 1/12*h^3*w, where h is the thickness of the beam, and w is the width.
So your maximum stress at the bottom of the beam would be 3*W*L/(2*h^2*w). For a laminate beam made up of six 19mm x 140mm boards stacked vertically on top of each other (5.5" wide, 4.5" thick), with a 150 lb gymnast in the middle of the 16' beam, your maximum stress at the bottom will be 2,681 kPa. That gives a safety factor of 22 (keep in mind that's for a static loading, not dynamic).
However, your stress at the bottom laminate surface (one board thickness up) will be 1,790 kPa, which is only a safety factor of 15, since the glue strength is the new weak point. I took a bit more time to research glulam properties (PDF), (after reading LeeG's answer) and thought about it some more, and realized I had oversimplified the problem. Wood is anisotropic with many defects, and the grain isn't always parallel to the board length. I also didn't account for the lumber and the glue to have different moduli of elasticity, which could allow one to stretch a bit and the other to take up some of the stress.
Glulam beams can also use stronger or flaw-free boards on the top and bottom (where the highest stresses are) to increase the overall beam capacity. A good analogy would be an I beam, with the web in the center mainly dealing with only shear loading. I also suspect glulam beams are laid up face to face, since it's much cheaper to use many narrow boards than a few very wide boards oriented vertically.
For the other option, turn the beam on its side, with the boards standing on edge, and the strength of the glue is no longer a factor. It merely holds the boards together for an additive effect from width, not a square effect from thickness. This puts less critical dependency on your glue up technique and the glue strength. Plus, it will be 5.5" thick, and 4.5" wide, which will be even stronger in bending.
So since you're dealing with a beam that's roughly the same dimensions in width and height (a big plus is the ~1" extra in thickness gained by standing them on edge), the boards you're using are all the same grade of lumber (I assume), and you may or may not have flaws in your glue up technique (no one's perfect), I'd stick with standing the boards on edge and gluing them side by side.
TL;DR: For this application, laminate boards standing on their edges, because science.