To what degree should a youth's home life influence the decision to declare him or her a delinquent?
I assume that you are asking about the degree to which the juvenile justice system should take a youth's home life into account when deciding whether or not to formally file a petition charging the youth with an offense. If so, I would argue that it should play a major decision.
The whole point of the juvenile justice system should be to try to make sure that youths do not end up becoming adult criminals. The system should do whatever is best for the youths--whatever will be most likely to help them be upstanding citizens as adults.
For this reason, the system should take home life into account. It should not formally bring youths from solid families into the system if it can help it. Such youths are fairly likely to be set straight by their families and would only be harmed by bringing them into the system.
This is, in a sense, unfair to those from bad home situations because it may rob them of their freedom while letting more privileged youths go free. However, it seems possible that contact with the system could help youths from bad home situations whereas it would possibly harm those from good homes.