Assuming most homes have "the box of instruments"!! (kazoos, wooden xylophones, maracas, recorders, harmonicas, awful electric organs, drums made from boxes, guitars made from elastic bands and shoe boxes.. ) what did they like?
Are they musical? If you put them in front of a piano and just get them to muck about, does it sound musical (not necessarily playing tunes)
Do they realise they will have to practise?
Are they performers or show offs?
If they want to play to get in an orchestra, then think about a slightly more unusual instrument such as cello, viola, double bass, trombone, cornet, baritone, French horn, percussion, saxophone also music scholarships at independent schools can be more valuable with more unusual instruments than the usual violin, flute, piano type instruments
If they are technical, then think about the guitar or piano.
How old are they? It is very important to start before they are 10 years old otherwise it will be more difficult as the musical brain is less malleable, but if they are willing to work, then any age is a good age to start.
They will have to learn to read music and music is very allied to language ability..
They have to enjoy it and you are paying for the lessons. Don't forget!!