File Manipulation in Linux

I want to search a text file for certain words and pipe them to
another file, each word on a seperate line. Here is an example:

File1.txt

index: A name_110022763763
index: A name_110022028309
index: A name_110022162534

I want to search File1.txt for all the name_xxxxxxxxxxxx words and
pipe them to another file each word on a seperate line. The other
file would look like this:

File2.txt

name_110022763763
name_110022028309
name_110022162534

Does anyone know how to do this?

Thanks

.