The laws on this will vary somewhat from state to state in the US. But in general, a person who convinces another to commit a murder might be convicted of conspiracy to commit murder, or accessory to murder. Being an accessory often carries the same penalties as being in principal, and in some jurisdictions there is no legal difference.
In addition such a person might be charged with a violation of Federal law, specifically 18 USC 373 - Solicitation to commit a crime of violence. Subsection (a) provides that:
(a) Whoever, with intent that another person engage in conduct constituting a felony that has as an element the use, attempted use, or threatened use of physical force against property or against the person of another in violation of the laws of the United States, and under circumstances strongly corroborative of that intent, solicits, commands, induces, or otherwise endeavors to persuade such other person to engage in such conduct, shall be imprisoned not more than one-half the maximum term of imprisonment or (notwithstanding section 3571) fined not more than one-half of the maximum fine prescribed for the punishment of the crime solicited, or both; or if the crime solicited is punishable by life imprisonment or death, shall be imprisoned for not more than twenty years.
In US law charges of "incitement" are limited by the "imminent lawless action" test laid down in Brandenburg v. Ohio, 395 U.S. 444 (1969) The court opion in that case states:
the constitutional guarantees of free speech and free press do not permit a State to forbid or proscribe advocacy of the use of force or of law violation except where such advocacy is directed to inciting or producing imminent lawless action and is likely to incite or produce such action.
This case held that a person may not be convicted of crime for speech advocating violence unless the advocacy is for immediate, present violence, or at least violence in the very near future, and there there must be significant likelihood of the violence actually occurring. This test has mostly been used in cases of public advocacy of violence to groups or crowds, but nothing prevents it from begin used in cases of one-on-one advocacy.
See also the Wikipedia article "Incitement".
I have not researched the specific charges in the Manson case.