Was Benjamin Franklin wrong (about conventional current)?

Electric current, A.K.A, "conventional current", is an abstract current, the flow of electric charge. From a previous answer I gave here:

Electric current is an abstract current, the flow of electric charge, not a physical current like, say, electron current, the flow of electrons.

But electric charge is a property of things, not a thing, i.e., electric charge is always "carried" by a thing.

So, while an electron current is necessarily an electric current (due the negative electric charge carried by the electron), an electric current is not necessarily an electron current.

For example, in a salt solution, we have two species of electrically charged ions present, the positively charged sodium ion and the negatively charged clorine ion. Imagine that the sodium ions are moving to the right and the chlorine ions are moving to the left.

Obviously, we have two ion currents in opposite directions but there is just one electric current and it must have a direction. The direction of electric current is, by convention, the direction of the flow of positive charge.

So, in this case, both ion currents contribute to an electric current to the right. The first term is due to the positive ions to the right. The second term is due to the negative ions to the left where the negative sign numerically "flips" the contribution to the electric current.

Think about it this way, if I told you that I was travelling at -60mph west, you'd know that I was actually going 60mph east. Similarly, a negative charge current leftward is an electric current rightward.

I don't think Franklin was "right" or "wrong", as it's just a choice of names.

As far as the particles are concerned, (to put it very roughly) we know one type of particle attracts another type of particle and repels it's own kind. We also know one type doesn't attract or repel itself or the others.
To distinguish between them and their properties, we call them something and say they have a certain type of charge - "Positive", "Negative" or "Neutral".

The electron is a lepton (type of fundamental particle) with a charge of -1e. e here is the unit of elementary charge. The proton has a charge of +1e, which is comprise of three quarks (two "up" and one "down") having a charge of +2/3, +2/3, -1/3 adding up to a total of +1.

Then everything else goes from here. As the link you give in your question says, we usually associate positive with "surplus" so it makes more sense for whatever has more of something to be the positive side. However, what Franklin had been calling "Positive" was the side with less electrons. Rather than swap the definitions round, they simply assigned the electrons a negative charge instead.

It's a bit like pipe with water flowing downwards through it - we say the current is in the direction the water is flowing. It would be confusing for may to say the current was flowing in the opposite direction, but this is how it is in electronics (i.e. we call the "water" negative) If we imagine the air bubbles flowing in the opposite direction, this is what we term "holes" (i.e. lack of an electron) and provide a mental image of the positive charge flow.
Of course, in a substances other than metal wires the current can be comprised of "real" positive particles or ions, as well as negative ones, so we can't always assume the current is an electron flow as Alfred mentions.