I have some ideas for the GOP appealing, but there is one that JC Watts said about why the GOP doesn't win the black vote is simply because they weren't showing up, look at the 2007 Morgan State debate where the top 4 candidates didn't show up to a debate on African-American issues (Watts also complained on how those candidates didn't have any black advisors). But this rings true to other groups, if you don't show up to talk about issues and offer your idea, people won't vote for you.
It may take me some time, but I have some ideas like I mentioned in the last paragraph on what they could do to win minorities.
Also from some articles I read, moderate to liberal Republicans get more significant percentages of minority voters than conservative Republicans do.
African Americans are a chicken/egg thing. I can see why they're considered a lost cause.
Hispanics and probably Asians are a different story. It's anecdotal, but I live around a fair number of high income 50s Asian/Indian families who have shifted Republican over the last decade as their incomes and taxes have gone up.
These people are really culturally conservative too. It sounds to me like an image problem, not a policy problem.