How do I ask a professor for a summer research project?
I get many letters of this kind, and I now summarily delete them without replying. Let me explain why. 


Money: it's never clear in these letters what kind of support the student wants/expects. The default assumption is that I would pay for them, and as an academic with limited resources, I have very little incentive to pay for an undergraduate to travel  from India to the US to work with me, especially when it's unlikely they'll be able to do much in three months. Which brings me to 
Project timeline: most letters of this kind are of the form "I'm interested in BROAD TOPIC A and want to work with you because you're interested in NARROW TOPIC B". For a three month internship to work, a project has to be very focused, and set from day 1. There's no time to explore. 
Self-interest: There's very little gain for me here, or at least none that's mentioned in the letters. If I want to hire an undergraduate, I can do it locally. I can even get financial support (sometimes) for doing that. I can't do that with a student from abroad, and I can't even vet them in advance. 


So any successful letter needs to address these three issues very effectively and quickly (because I delete these emails without reading more than a few lines). 

Having said that, I know of at least one example where a student came from India to work with a colleague and that summer project turned into an application to grad school. The student is now at my university.