I disagree with the other responses that have really keyed in on the father/child relationship here. Your story would not have been extraordinary or exceptional at all if you had omitted the family relationship entirely. "I'm an exceptional worker vital to my business but I'm underpaid and can't get a raise. I would consider leaving, but I like my job and don't want to quit." That sums up the daily thoughts of a huge segment of the working population.
You like your job and you don't want to quit. Sure you might make more money some other place, but what would you be trading for that money? Would you have to move somewhere you'd rather not? What kind of manager would you have? What would your career path look like? How much job security would you have when the next recession hits, compared to your current job? What's worth more to you, the extra short-term cash or avoiding some of those likely unfavorable changes? That's all before considering the family angle at all, which provides benefits on the career side and possibly outside of your career as well.
I think that the advice I would give if this were NOT a family business would be to talk with your supervisor or a mentor within your company about your career goals and aspirations, about your current frustrations, and your perception of how you're doing. See if you need a reality check on your self-perception, and try to get some solid advice about whether you're on track to achieve what you'd like in the longer term, or (at least implicitly) whether a career move would make you more likely to realize those goals. This advice is a little bit hard to implement in most cases, because those conversations can be very sensitive and with the wrong boss and wrong situation could cause your situation to deteriorate. However, having that conversation with your own father is a luxury! I think the advice holds even more within a family business than it would in the corporate world.
Good luck
-- JC