Why did George H. W. Bush select Dan Quayle as his running mate in 1988?

This has not been a usual observation, but Quayle actually brought considerable strength to the ticket through his family's conservative Pulliam news dynasty. Quayle's mother was a Pulliam, and the family controlled the editorial content of newspapers throughout the Midwest and Southwest. Ever wonder why Indiana and Arizona are nearly always bonded together in a conservative political lock-step? The Pulliams' powerful enduring influence is a major reason for this.

Bush's introductory speech on Quayle began, "Born in the middle of the country, born in the middle of the century..."

I believe that to George Bush Sr. (an aristocrat), the circumstances of Quayle's birth, background and breeding seemed to outweigh his other qualities. Put another way, it seems that Quayle was selected for "social" reasons of personal chemistry rather than political considerations. As J.P. Morgan said, "There are two reasons a man does something; the "official" reason and the real reason."

This is coming from someone who has voted for every Republican presidential candidate not named Bush. In voting for Dukakis, I voted for "competence, not ideology," and the proposition that the son of immigrants like him (or yours truly) might possibly become President of the United States.

Although it must be admitted that the Bentsen-Quayle matchup played a role in my decision.

Although Quayle is the punch lines of many jokes, he had served with distinction, and had been elected with significant margins; those are strong positives for a Vice Presidential Candidate.

He won reelection in 1978 by the greatest percentage margin achieved to date in that northeast Indiana district. In 1980, at age 33, Quayle became the youngest person ever elected to the Senate from the state of Indiana, defeating three-term incumbent Democrat Birch Bayh by 54%-to-46%. Making Indiana political history again, Quayle was reelected to the Senate in 1986 with the largest margin ever achieved to that date by a candidate in a statewide Indiana race, easily defeating his Democratic opponent, Jill Long with 61%. His 1986 victory was notable because several other Republican Senators elected in 1980 were not returned to office. Wikipedia

As noted by others, he appealed to one of the core Republican demographics (fundamentalists).

Wikipedia doesn't note his specific expertise; Senator Quayle had served on the Arms Control and Disarmament committee and from what I understand was perceived as experienced and educated in that field.

During his tenure in the U.S. Senate, Mr. Quayle became widely known for his legislative work in the areas of defense, arms control, labor, and human resources. With his service on the Armed Services Committee, the Budget Committee, and the Labor and Human Resources Committee, he became an effective Senator, respected by colleagues on both sides of the aisle. biographybase

Arms control is one of the areas in which Dan Quayle has developed significant expertise. He took a particularly close interest in the negotiations leading to the treaty banning intermediate range missiles. R. Perle in NY Times Perle spent much of his career on arms control, and his opinion should not be taken lightly.

. . . Quayle involved himself in foreign policy issues through the Armed Services Committee. As a freshman, he took the lead in persuading other freshmen Republicans to reach a compromise on a Reagan administration plan to sell AWACS surveillance planes to Saudi Arabia. Quayle arranged for Reagan to sign a "letter of certification" that satisfied enough otherwise doubtful senators to win approval for the sale. Quayle was also willing to take positions independent of the administration. In 1987, as the Intermediate Nuclear Forces (INF) Treaty moved toward completion, Quayle joined a group of conservative Republican senators in opposition. When President Reagan accused them of accepting the inevitability of war, Quayle denounced the president's comments as "totally irresponsible." A question arose over whether the treaty covered such "futuristic" weapons as lasers, particle beams and microwaves. Both the State Department and the Soviets agreed they were covered, but Quayle insisted they were not. (Later it became evident that the economic deterioration of the Soviet Union severely hampered its ability to compete with the United States in developing such sophisticated space weapons.) "Senator Quayle came at me repeatedly with complaints about this issue," Secretary of State George Shultz recalled. At last the secretary begged, "Dan, you have to shut down! We can't have the president's achievement wrecked by Republicans!" The treaty was finally approved by a vote of 93 to 5, with Quayle voting in favor. Senate History which has a very nice summary of his career; highly recommended

At the time, Arms Control was a significant national issue; today's problems blind us to how important we perceived the issue at the time.

In short the man had the ability to deliver votes, the ability to deliver votes from a key constituency, and some experience to bring to the ticket. He wasn't an entirely irrational choice given what was known at the time.