Are there "restart" mechanisms for candidates seeking a graduate admission with a mediocre undergraduate record? To take one example, the University of Pittsburgh offers certificate programs equivalent to a minor to non-degree students, and also offers a chance at a "second" bachelor's degree.

Suppose there was an undergraduate in say Mathematics, who got at best mediocre grades, no more than a 3.0 GPA in math, and a 3.0 for the degree. I would imagine that this person would not a be a good graduate school candidate for a program in say, Computer Science.

Suppose some years later, the same person "woke up," enrolled in a certificate program in Computer Science, equivalent to a minor, or better yet, took a whole second degree (equivalent to a major), and got (mostly) A's, say a 3.8 or better GPA. The new program includes As in one or more graduate courses. The person's GRE scores have also climbed, to over 700 in each category. This person then wants to do graduate work in Computer Science.

Would this cause some graduate school programs to take a "fresh look" and him or her graduate student credentials? Perhaps this is most likely to take placein the same school where s/he successfully completed the certificate or second degree program?  That is to say, is the strength in the certificate program or second major program enough to call into question the mediocre undergraduate record? (The person's letters of recommendation would come from professors in the later program.) Put another way, is there a point where a strong enough "post graduate" record will make up for a mediocre undergraduate record?
I don't think it is possible to give a general answer to this question. There are too many variables. On one hand people do recognize growth in a student and that new work may be more important than old. However, it also depends on what you mean by a "certificate program". There is a lot of variability in that. Some have good reputations, others not. How does a grade in a given certificate program translate to a grade in a university course? How much skill was gained in the course. 

Yet another variable is which courses were taken. Is it more or less the equivalent of an undergraduate major, including all of the usual courses. 

Given all of these variables, the only answer that is valid will come from a particular institution evaluating the candidates complete record and their statements about themselves and their goals. For some, the answer would be no, for others, possibly yes. The only way to know is to ask and the best way to ask is to actually apply and supply all of the needed information, including reasons why the record is a bit "unusual."