A raise is, by design, a permanent change to your compensation. Staying with relatives when you travel is a temporary cost savings; it's only a factor so long as (a) they continue sending you there and (b) your relatives continue to offer you accommodations. Asking for a permanent increase in compensation for a temporary reason is almost always a bad idea in my experience.
What you can do is to ask for small one-time "bonuses", not necessarily cash. I once stayed with friends during a conference and asked for a couple extra vacation days (treated as comp time, not a permanent increase in PTO) in exchange, pointing out that I'd saved them $X and the "cost" (daily salary) of the vacation days was <$X. They went along with that, and because we didn't have to submit timesheets my manager was able to just say "do it" and not do any paperwork.
You'll note the use of "<" in "<$X" above. I've found it a good practice, when asking for informal compensation, to always ask for a little less than what the strict number-crunching suggests. If you seem to be accounting for things down to the penny, it can raise concerns about what else you're being that precise about -- for example, are you leaving when you've logged exactly 40 hours for the week, even if another hour or two would make a big difference to somebody? Don't raise those doubts if you're salaried (rather than paid by the hour).