Another PhD student is going to present my research I'm doing a PhD with a professor with serious funding problems. He invited me to prepare a presentation for a conference, making it clear that he wouldn't pay for my travel expenses. I, in turn, replied that I didn't have the money to travel to that conference. He offered that another student go present my research because that student, unlike me, is ready to pay for the travel expenses.

My professor claims it to be common practice, but I'm having a hard time wrapping my head around the fact that a student who didn't contribute the slightest bit to my research will be presenting my results. Is it really common practice? What do you think about that offer?
In my experience, it is very common for one person to present someone else's conference talk.  I know this happens because at about 5% of talks speaker announces they are doing it at the beginning of the talk.  The reason for this occurring is not usually given.  I would guess the most common reason is that the original speaker changed jobs or graduated, and is no longer able to attend.  I suspect that some speakers wish to give more talks (usually there is a limit of one or two per speaker), so they ask their students who are not able to attend to submit talks for them, and then change the speaker at the last minute.

Personally, I have given talks for colleagues twice.  The first time, the colleague was having a baby (it's mysterious why he didn't know this until a month before the conference) and I gave his talk.  In the second case, the colleague quit science altogether well before the conference.  We asked the conference to remove his abstract and put a new one, with a new author list, in its place.  I made the talk from scratch, with limited content about his work.

I think your professor's proposal is reasonable, but that it would also be reasonable to refuse or to specify how you would like to be credited.  But in my field, conference talks have near zero prestige.  Computer scientists see things differently.