I agree with Michael, neither of those is good for this request. However, as to the difference - "Lend me a hand" is for when you want someone to assist you with something right at that moment. Whatever the activity is, you're doing it (or about to do it) and they will help you get it done better or faster because the two of you will be working on it together. Think of it as "Can you do ________ with me, right now." (e.g., "Can you lend me a hand and help me carry this chair?") "Do me a favor," is when you want someone to perform a task that will help you. It usually has some level of significance. It doesn't have to be done right away (though it may be) It may be something that you can't do yourself (again, it may be). (e.g., "Could you do me a favor? Can you ask your mom for her turkey recipe?") The other thing is it does create a small sense of obligation - if someone does a favor for you, you owe them a similar level of favor if they ask. There is overlap between the two things, but they're slightly different. Either way, though, a reference request from a professor doesn't fall in either category. The person isn't writing a reference with you, so they're not lending you a hand. And while a reference will help you - it's not really the type of help that's a "favor."

I don't recommend either of them. Perhaps: "Request for a reference letter."

"Give me a hand" and "do me a favor" are both informal ways to request for someone's help, and it's something I don't recommend that you say to your professor. If I were in your professor's shoes, I'd feel slighted at a rudeness of it. Obviously, this isn't your intention, so I'm not going to dwell on it. From my experience a few years ago with requesting for reference letter, I have said something along the lines of "I am hoping to ask for a reference letter from you." Be humble (it's your professor who is doing this as a favor to you, not the other way around), and be sure to express why you chose them to write this. Good luck!