In order to get a sustained current, you need to put electrons in one end and take them out from the other in equal numbers. Thus mass doesn't change.
However, if you're talking about static electricty, you can actually create a mass imbalance by taking a few electrons away from one side or putting some electrons on the other. It's a very, very, very small change in mass.
Whoa, really? Their answer was pretty spot on to your question. Why take such a hostile attitude when it’s your own inability to understand? I’m shocked to see such an ignorant and condescending comment being upvoted.
There’s nothing wrong with not understanding a technical explanation and asking for it to be simplified to your level of understanding, but this is not a simple concept to explain because it is ACTUALLY complex and counterintuitive. How is it their fault?
I'm just trying to answer your question, but I don't think you're exactly clear on what you want to know. It's ok, no need to get defensive about it.
Whether or not 'one end of the wire gets heavier' depends on what your doing. If you are using the wire to power an LED from a battery, then no, because electrons are removed from one end and placed into the other at equal rates. Charge and mass within the wire are both globally and locally conserved.
If you do something where charge/mass isn't conserved such as removing electrons from one side (i.e. by rubbing a fork on a carpet) or by using an electric field to 'tilt' the electrons to one side, you can create a (very small) mass imbalance. This activities are not usually considered to be useful electrical current.
You're talking about conventional current, and it's nothing more than a polite fiction. We model circuits as current moving from positive to negative, but with the implicit understanding that the real charge moves the opposite direction.
Positive charge carriers do not actually exist[0]. There's only electrons and holes they can go into. We can talk about the movement of holes, but that's a virtual charge carrier at best.
Conventional current is just a convention. It's what we started with (because Franklin was wrong) and it's too much effort to change now. In practice, the distinction almost never matters. Sometimes it does, but not enough that it's worth overhauling the entire field of electronics.
[0] of course positrons and protons exist, but they aren't relevant to electronic circuits
