Lady Jane Grey lost her throne due to the machinations of Privy Council factions as much as anything else, as I understand it, but there did seem to be general public rejoicing at the accession of Mary, even though most of the people must surely have known she would try and reconvert England.
Was it purely a case of "Jane just doesn't have royal blood"? Would people really welcome a "bad" queen over a "good" one, just because the "bad" one had the right genes?
You can talk more broadly about the subject of the "rightful" monarch if you wish. It's something I've noticed in a couple of famous medieval-based fantasy series; for Tolkien, the royal blood is everything. Aragorn just rocks up at the royal palace at Gondor, flourishes his sword and everyone instantly rejoices at the return of the king, despite them having gotten along quite nicely without one for centuries. The only dissenters are the evil, power-hungry current regent, Denethor, and his overly-proud son Boromir.
Move to the more modern A Song of Ice and Fire, and the common folk don't care two beans who's king or queen as long as the harvest comes in and they don't get their house burned down by one of the warring factions. Gossip about the goings-on at court is treated more like Hello! Magazine than important matters of state, and some probably don't even know or care who the king is.
I wrote about the history of England's Queens here, with a mention of Lady Jane Grey (in pt 2)
In essence it came down to suspicions that Lady Jane Grey was being used as a vessel for her male relatives to obtain power by stealth and to the fact that despite the official separation of England from the Catholic Church in Rome there were still plenty of Catholic sympathies in the English Parliament and the English army so Mary was able to garner ample support for her cause.
Jane did have royal blood and had a legitimate claim to the throne. She was already in the line of succession before Edward made her his heir.
Jane's younger sister, Mary Grey was actually Queen Elizabeth's heir presumptive until Mary Grey's death.
As for Queen Mary. She didn't start out bloody or bad. She was loved by the English people and the beginning of her "bloody Reign" didn't really start until Philip II refused to marry her with Jane still living. It was a condition of their marriage that Jane be executed. Mary was in love with Philip and was afraid to loose him. Then she began her real persecution of "heretics" who failed to convert to Catholicism. She supposedly believed that God was punishing her with her "failed/hysterical pregnancies" and ramped up the executions to satisfy God in hopes for an heir.