The correct option is
B to read
The given sentence is complete in meaning, with the subject (she) + verb (gave) + indirect object (me) + direct object (very nice book).
The blank that follows is likely an object complement. Object complements are expressions (nouns, pronouns, adjectives) that tell us more about the direct object.
Option C: 'To read' is an infinitive (also a noun phrase). Infinitives function as object complements, among other things. It complements the direct object 'book', and hence is the most suitable option.
Option A: 'Reading' is a gerund. Gerunds are noun forms (verb + ing) that function as nouns. But here, it cannot complement the object 'book'.
Gerunds express actions that are real or have been completed, whereas infinitives express actions that are unreal or are yet to happen.
In the given sentence, the action has not taken place yet. Hence option A is incorrect.
Option B: 'Read' is a verb and verbs do not complement objects.
Option D: 'Will read' is also a verb phrase and does not complement the object.
Hence options B and D are wrong.