Does an elevator gain weight while it lifts something?

No. Weight W = mg is based solely on the mass, m, and gravity field strength, g. Those do not change, no matter which way the lift goes.

But what does change is the effective or net weight w = W +/- F = m(g +/- a) where F = ma is the force added or subtracted due to the elevator accelerating upward (added) or downward (subtracted).

So in your case, there will be a brief "gain" in net weight while the elevator accelerates upward. But once the lift is no longer accelerating, that gain, F = ma, disappears and w = W, which is just the real weight. F = ma disappears because the acceleration disappears and we have a = 0; so F = ma = m0 = 0.

Note this is true with or without friction. As long as the lift accelerates, either direction, the real weight W will be added or subtracted by F = ma depending on which way the lift is going as it accelerates.

Does it gain weight? Well, no. The weight (act of gravity on a mass) of an elevator will increase if it were to have an object (such as a human) placed on/in it. However, as the elevator is moving, it is the Force of gravity (Fg= m*g) and Force of Friction (Ff= mu*FN) that act on the moving object. It may gain weight if collecting mass on the way up. That's about it.

Take care.

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No it doesnt