Why is H⁺ ion considered the source of acidity, and OH⁻ considered the source of basicity?
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.