Well, there is definitely no direct regular transport (neither bus or train), so at least one change will be required. Also it might be not very fast, since though distance between Martin and Budapest is not very essential, but transport routes aren't really straight in terms of geography.
Taking into consideration arrival time (9:50am), it will be nice to catch 11:25 train from Budapest to Bratislava (arr.14:06). Hour and a half is enough to make it from airport to Budapest-Keleti train station (trains to Bratislava depart from there), in order to do so your friend will need to catch bus #200 from airport to the nearest metro station (Kőbánya-Kispest), buses run every 10 min. at that hour of the day, then take metro to Keleti station. Train ticket Budapest-Bratislava-Budapest cost 17 euro (valid for 1 month, so can be used on the way back). While from Bratislava your friend has two major options: a) catch direct bus Bratislava-Martin (nearest will depart at 15:20), takes roughly 4 hours and cost 10,7 euro or b) continue by rail with change in Zilina and Vrutky, basically same travel time and fare as the bus.
Regarding way back it will be pretty much the same thing, but in reverse. You friend will have to start from Martin around 10am in order to make to Budapest till 18-19. In total it will be possible to travel Budapest-Martin-Budapest for less than 40 euro
Also you might find some ideas in the similar discussions we had on forum some months ago, route was the same -
http://europetrainsguide.com/forum/index.php/topic,502.0.html (here you also find all the links for timetables, both train & bus)
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