Okay, here's a quick and dirty design of steel sistering for a floor joist.
The measure of the "stiffness" of a beam in bending is it's section modulus (S). That is a function of the beam shape, but for a floor joist is width x height cubed divided by 12 and divided by half the height. For a 2x10 that number is 21.4.
A steel C channel, 3 inches wide x 4.1 lbs/foot (C3x4.1) has a section modulus of 1.1 when used upright.
Steel is 45 times as strong as wood in bending. So the C channel has an equivalent strength of 45 x 1.1 = 49.50 compared to the wood strength of 21.4
So, if you sister a 3" C channel weighing 4.1 lbs/ft to the floor joist, it will be approximately twice as strong as the original joist.
To do that, I'd put a section at least 4 feet long, and drill through it at each end, and at 6 inch increments along the length. Put a 1/4" bolt through the joist and the sister C channel at each hole, and clamp it tight with a large flat fender washer on the wood to keep it from crushing under the nut there.
Longer sections just reinforce it more, but 4 foot is more than adequate to bridge across a plumbing cutout. The attachment to the wood joist will be adequate to keep the steel from kinking and twisting under load, so it won't fail by that mechanism.
Now, to cut the calculation closer: 21.4 divided by 45 = .48 = required section modulus of a steel section to equal a 2x10. A 3" x 1/4" steel bar has a section modulus of .56, or slightly more than the equivalent to a 2x10 wood joist. In theory you could bolt a 3" x 1/4" wide plate to the joist and have an equivalent sister. It might fail by kinking and bending sideways. A 3" angle iron with 1/4" thick legs would be a better choice, similar stiffness but won't kink sideways.
So, two good options; a 3" x 1/4" equal leg angle or a C3x4.1 channel will sister a 2x10 safely. A probably acceptable option would be a 3" x 1/4" bar.