Let's examine some laws that may apply to your case
Digital Millennium Copyright Act (DMCA)
There are criminal penalties for willful infringement for personal financial gain. If you are only sending it to one family member through a private communication (eg not posting it publicly) and not selling it and one copy of the music video has a retail value of $1,000 or less then there will be no criminal penalties. But you may be liable for civil penalties.
Computer Fraud and Abuse Act (CFAA)
Courts have ruled that ToS violations do not constitute unauthorized access (or exceeding access) so you aren't in violation of the CFAA
Civil Liability
By downloading a video off of YouTube you are in violation of their ToS. There isn't enough prior case law for me to tell you the outcome here. You may be committing some sort of civil wrong by the act of downloading. In addition, you are distributing a copyrighted work without permission which would open you to more civil penalties.
The real question is how likely is it for this to be enforced? Not likely.