During President Trump's impeachment trial the White House counsel have sought bringing Hunter Biden in several times for testimony.
How is Hunter Biden's testimony relevant, opposed to that of John Bolton and others witnesses?
'Why' questions are inherently difficult, often de-evolving to opinion-mongering. Unless someone in the White House tells us their reasoning explicitly, we could only guess.
However, what we can say is that the White House and its supporters have consistently held that the Bidens and Burisma were involved in some unspecified form of corruption in the Ukraine, and that this justifies all of Trump's activities. I use the term 'unspecified' here because various suggestions about that corruption have been tossed into the ring at various times, but all such suggestions have been (shall we say) inconsistent with observed facts. In any case, calling Hunter Biden as a witness would be consistent with the administration's position. If they are really trying to argue that Trump's quid pro quo and obstruction actions are appropriate or excusable because of Hunter Biden's corruption, then calling Hunter as a witness to expose this unspecified corruption would be a natural move.
Whether that is a valid legal strategy is a different matter; "He did it first" rarely flies well in a real court of law...
It's worth saying that calling Hunter Biden (regardless of his guilt or innocence) as a witness serves the purpose of slandering the Bidens as effectively as having the Ukrainian judicial system announce an investigation. If we assume Trump was trying to play that card originally, then we can assume that he is still trying to play it.
Whataboutism. It's easy to make potshots at vague 'questionable' behavior without actually trying to get to the bottom of it and punish those responsible. Add to that a refusal to defend one's own actions, and people quickly come to believe that 'all politicians do (X bad behavior)'. This has worked amazingly well for Trump against Hillary Clinton, and is working just as well for Biden.
Normalizing corruption. If they can prove that Hunter Biden did anything wrong at Burisma and then prove that Joe Biden did anything wrong by pressuring Ukraine to get rid of the guy who wasn't punishing Burisma fast enough, then both parties are bad, and people need to re-calibrate their expectations of how politicians should act.
Making vigilantism sexy. Once they've muddied the waters and made people believe that all politicians are the same, they can work on portraying Trump not as a law-breaker or someone trying to enrich himself at the taxpayer's expense, but as a vigilante, someone who doesn't follow the rules but gets the job done; an unorthodox but effective swamp-drainer.
Turnabout is fair play. The Republicans haven't gotten much traction out of investigating the parties involved in the Mueller investigation, probably because Mueller's team was insanely professional and voters see these investigations as partisan. However, if Joe Biden is guilty of something, then they can comb through his (prodigious) background and cast aspersions on his patriotism via endless corruption hearings, and thus take back the narrative from the Democrats.
Trump has been credibly accused of abuse of power, solicitation of a bribe, and obstruction - all in order to (dis)inform the US voters that Joe Biden may be corrupt.
Of all the alleged crimes to be judged in the impeachment trial, creating the public impression that Joe Biden is corrupt was the alleged payoff, the purpose, the motivation.
Since there has been no indication that Hunter Biden possesses relevant information about the abuse of power, solicitation of a bribe, and obstruction this impeachment is about, the most logical reason that remains is that Trump's Republican supporters want to call Hunter Biden because they want to do the exact same thing Trump does: to create the public impression that Joe Biden is corrupt.
To summarize comments that have since been nuked: Their point was that if an investigation into Hunter Biden were to turn up evidence of wrongdoing by either Biden, that evidence can then be retroactively serve as Trump's motivation for asking the Ukraine to announce an investigation, which would legitimize Trump's actions. My counter argument is that neither the law nor logic work that way, because real life events happen in chronological order.
One of the criticisms of the alleged quid pro quo is that Trump's request was purely political, calling on a foreign power solely to manufacture or dig up dirt on a sitting president's opposing candidate.
However, if it can be shown that there was reasonable evidence to suspect foul play in which Joe Biden may have been involved, the grounds for impeachment become murkier, because this would be a legitimate investigation - a candidate who may have personally benefited from his previous role as Vice President through corrupt dealings.
As the facts stand now, Hunter Biden had no qualifications related to serving on the board of a petroleum company, and exactly what he was being paid his large salary for has not been made clear. Moreover, it is also an established fact that former VP Joe Biden bragged about threatening to withhold some $1 billion in aid until an allegedly corrupt Ukrainian prosecutor was ousted - and supposedly this prosecutor has just begun an investigation of Burisma. This could have been interpreted as similarly abusing power of office for personal benefit, thereby justifying a corruption investigation of not just a current candidate, but a former VP.
Kinda surprised at the answers so far, which seem to miss the point. Republicans think Hunter Biden's testimony is relevant because:
“I told them, look, nothing matters more than the facts on Burisma,” Sen. Cruz (R-Tex.) told conservative radio host Hugh Hewitt on Friday, adding: “They built their entire case on this house of cards. Lay out substantive, factual reasons why investigating Burisma — the president had a responsibility to do so.”
Note Republicans aren't the only ones to think this is relevant. At least one Democratic Senator has said the same (source).
Calling Hunter would be partly meant to change the subject, to put the Bidens on trial rather than Trump. It would also be an attempt to extract political payback from Democrats for daring to try to hold Trump accountable.
For now, though, it’s mainly a threat designed to get Democrats to back down. Senate Majority Leader Mitch McConnell has privately referred to the witness issue as “mutually assured destruction.” In large part, that’s because of Hunter. Republicans believe that any testimony from Hunter could be tremendously ugly for Joe Biden’s presidential campaign — so ugly that Democrats would never permit it to happen.
(Same source as above)
Per Pam Bondi, the democrats brought up Burisma/Hunter Biden over 400 times. So the republicans felt obliged to respond. Source: https://www.nbcnews.com/politics/trump-impeachment-inquiry/live-blog/trump-impeachment-trial-live-coverage-president-s-defense-begins-day-n1123371/ncrd1124211#liveBlogHeader