Of course, You can make engraving just "Тебе" or "Для тебя" but it'll be unusual and unnatural for inscription like these in Russian. Usually such inscription content an indication of the recipient of the gift and the event in honor of which the gift is being given. For example "Другу в день рождения". "Дорогому отцу в честь юбилея" If you are gift smth. to smb.just because without any event, the sign may be "На память от (your name)". Other variant "На добрую память" The specific inscription depends on whom you give the gift to and why.

Я хочу отправить подарок (send it to you) тебе or Я купил подарок (do it for you) для тебя

Hum, the problem is that it isn't.. not something we are used to seeing engraved on gifts or written on envelopes:-) I think, "for you" is customary in English speaking countries... But I think not "to you", right? If I'm right, then may be you would see what I mean if you imagine a gift inscribed with "to you". Here both are uncommon, so whatever you choose, a Russian would read it with 'fresh eyes'. You are 'inventing' a new Russian expression. Both тебе and для тебя are grammaticaly correct. You may think of writing in English or writing a different text. On the other hand, тебе or для тебя may look a bit odd, but 'odd' is not 'bad': they are also less boring:) And it is hard to choose among them:( In general, 'тебе' is more similar to 'to you' (or simply you, her, etc.) and 'для тебя' is more like 'for you', but there are situations when you have 'for you' in English and we say 'тебе' in Russian.

Yes, "for you" is all I want to write - would the dative (e.g. "тебе" or "мою другу") be better? Thanks for your answers!

hum, if "for you" is the WHOLE message, then it is not very common an inscription on gifts in Russia.