"What is the sign for the little preposition 'for' in American Sign Language?" Like other prepositions, ASL grammar is much involved that the sign 'for' may be used differently or not used in sentences. If you're a brand new learner, ASL is a complete language of its own. Same true for other different sign languages.
Meaning: Intended to belong to, or be used in connection with.
Pronunciation/articulation: Dominant "1" handshape or index finger is in contact with the temple of the head and the palm orientation faces inward. The forefinger moves outward away from the head in an arc motion where the palm orientation faces out.
Usage examples: "for who/whom?", "You don't want it? I bought it for you yesterday!"
The wh-question glossed as "FOR++" or "FOR-FOR?" with the facial expression (including furrowed eyebrows) is a synonym for "WHY", but it's not synonymous with "Why?". Deaf people used it with a specific meaning in contexts. This single "FOR-FOR" question can function as "What is it for?", "Why?" (depends!).
For beginners or learners, it's safe to use WHY instead. If using it incorrectly in the slightest way, it can convey a different meaning, like inquiring or investigating someone "Why are you (insert here)?" So much nuances in this natural language of the Deaf native signers.
Handshape minimal pairs for FOR: DONT-KNOW, SMART.
Dominant-handed handshape rhymes: PAH (SUCCEED), NOT-MIND, CHECK.
First 100 words.
As you feel more comfortable with the first few hundreds of ASL signs, progress further with your vocabulary and learn signing more.