Question

insert a correct participle. Look , my grandmother has __________(weave) a scarf for me!

Answer 1

I'm afraid the question is wrong as has woven in this sentence serves as main verb in present perfect tense. There is no participle in this sentence in the context you have given. 

If you rephrase the sentence as - My grandmother has a _______ scarf for me. - then you can have woven as a participle there.

(A participle is a verb with the force of an adjective.) 
So, woven becomes an adjective that describes the noun i.e. scarf.