Yours is a historical question. Note that in modern chemistry, it is not required that acids have H+ or bases have OH-. The words acidity or basicity referred to certain common behaviors such as acids will liberate hydrogen when come in contact with metals, or they will decompose carbonates to carbon dioxide. Similarly, bases had other common characters. Indicators behaved the same way with acids or bases. No one knows why bases were called bases. Acids we understand, that they tasted sour (=Latin). The term alkali (Group I bases) makes more sense (from Arabic for soda ash or potash).
Definitions are made by humans and I am glad you are not blindly accepting the definitions and at least thinking about them. Arrhenius was among the first ones to think why solutions conduct electricity. When Arrhenius proposed this idea in his PhD, but he knew that his mediocre committee members will not like it. How can anything ionize in water? It was a stupid idea to them. It turned out that most common acids and bases had very high conductivity better than anything else known, even better than salt solutions. Say 0.1 M HCl had way higher electrical conductivity than 0.1 M NaCl? Why? The "proton" in aqueous solution has the highest conductivity known today. What comes next? The hydroxide ion. Also in Arrhenius' time electrical decomposition was known. Faraday did a lot of electrochemical experiments then.
When you electrolyze an acid solution, what do you get at the electrode which has a negative electrostatic charge. Hydrogen gas! What appears at the positive electrode, chlorine gas. How can "H" and "Cl" reach the electrode, this is possible when they have a charge on them as well. So, "H" in acids must exist as "H+"; as a positively charged entity. I have not done justice to Arrhenius history or the history of acid bases.
If you are really (or better word genuinely) interested then search the Theory of Ionization+ Arrhenius History in Google Scholar. Partington's History of Chemistry will have it too but it is meant for serious scholars of chem. history (multiple volumes). It is available online from the Internet Archive.
Answer from ACR on Stack ExchangeYours is a historical question. Note that in modern chemistry, it is not required that acids have H+ or bases have OH-. The words acidity or basicity referred to certain common behaviors such as acids will liberate hydrogen when come in contact with metals, or they will decompose carbonates to carbon dioxide. Similarly, bases had other common characters. Indicators behaved the same way with acids or bases. No one knows why bases were called bases. Acids we understand, that they tasted sour (=Latin). The term alkali (Group I bases) makes more sense (from Arabic for soda ash or potash).
Definitions are made by humans and I am glad you are not blindly accepting the definitions and at least thinking about them. Arrhenius was among the first ones to think why solutions conduct electricity. When Arrhenius proposed this idea in his PhD, but he knew that his mediocre committee members will not like it. How can anything ionize in water? It was a stupid idea to them. It turned out that most common acids and bases had very high conductivity better than anything else known, even better than salt solutions. Say 0.1 M HCl had way higher electrical conductivity than 0.1 M NaCl? Why? The "proton" in aqueous solution has the highest conductivity known today. What comes next? The hydroxide ion. Also in Arrhenius' time electrical decomposition was known. Faraday did a lot of electrochemical experiments then.
When you electrolyze an acid solution, what do you get at the electrode which has a negative electrostatic charge. Hydrogen gas! What appears at the positive electrode, chlorine gas. How can "H" and "Cl" reach the electrode, this is possible when they have a charge on them as well. So, "H" in acids must exist as "H+"; as a positively charged entity. I have not done justice to Arrhenius history or the history of acid bases.
If you are really (or better word genuinely) interested then search the Theory of Ionization+ Arrhenius History in Google Scholar. Partington's History of Chemistry will have it too but it is meant for serious scholars of chem. history (multiple volumes). It is available online from the Internet Archive.
Aqueous solutions
In aqueous solution, 
and 
are always present because of the autoionization of water:
So in aqueous solution, I would be comfortable saying there is an "association between 
and acidity, and between 
and basicity". When the two concentration are equal, the solution is neutral. When the concentration of 
is larger, the solution is acidic. Finally, when the concentration of 
is larger, the solution is basic.
This is just the definition of acidic and basic aqueous solutions, and can be used to define acids and bases in an empirical way (i.e. not knowing the underlying details of atomic structure and chemical bonding). The chemistry of aqueous solutions often is vastly different depending on whether they are acidic, neutral and basic, so it makes sense to introduce these concepts (and even have a quantitative measure for them, the pH of the solution).
Arrhenius vs. Brønsted-Lowry definition
The hydrogen ion plays a role in defining an acid both according to Arrhenius and Brønsted-Lowry. It seems that in defining a base, however, that the hydroxide ion is required according to Arrhenius and not mentioned according to Brønsted-Lowry. If you combined water autoionization with the Brønsted-Lowry definition, you get the hydroxide ion back. So why does the definition of acid stay the same and the definition of base changes or suddenly apply to different substance. In other words, is sodium hydroxide just an Arrhenius base and ammonia just a Brønsted-Lowry base (see an answer here: Is Sodium Hydroxide a Bronsted-Lowry Base?)?
Ionization vs. dissociation
Some substances, such as 
or 
ionize when they react with water, either giving off a hydrogen ion or picking up a hydrogen ion. Ionization is a good term for that process because neutral species react to form ionic species.
When bound to these species, the hydrogen atom is connected via a covalent bond. There is no ionic substance I know of that contains 
-ions. When we write 
, we imply that the hydrogen ion is covalently bound to a water molecule (
) or a cluster of water molecules.
On the other hand, there are ionic species that contain 
ions, for example 
or 
. When these are added to water, they dissolve into solvated ions, e.g.
This process could be described as dissociation, and does not look like it involves hydrogen ions at all. If you isotopically labeled the sodium hydrozide, however, you would realize that they mostly react to form water, as in:
Why is H⁺ ion considered the source of acidity, and OH⁻ considered the source of basicity?
It is not. The source of acidity is a substance that has an ionizable covalent bond to hydrogen. The source of basicity is a substance that can form a covalent bond with hydrogen by reacting with a hydrogen ion. This includes hydroxide ions. In aqueous solution, the consequence of adding acids or bases to aqueous solutions is a increase of the concentration of hydrogen ions or hydroxide ions (with a decrease in hydroxide or hydrogen ions, respectively). In non-aqueous systems, where there might not be any source of hydroxide ions or they would not be solvated, the acid/base concept is limited to hydrogen ion transfer, or expanded to consider a different anion (such as 
in liquid ammonia).
Can we make this easier to learn?
It is a conceptual hurdle to introduce 
and 
as the first examples of acids and bases because they are so different from each other. It would be easier to start with acetic acid and sodium acetate, or ammonia and ammonium chloride. A nice pair of strong acid and strong base would be 
and 
, or 
and 
.
Once acid/base concepts have settled in, the somewhat weird case of 
could be introduced, along with mentioning that while this is an example of an ionic substance that already contains hydroxide as an anion, no comparable ionic substance containing hydrogen ions as a cation exist.
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What about acids and basis who don't have hydrogen or hydoxid ions in them, how would we classify them?