His parents lost their first son so they brought him up with extra care, love and attention
His parents lost their first son so they focused all their love, care and attention on their remaining son
「育てられる」= be brought up / be raised
「大事に」= with great care / with extra love and attention
「一度男の子を亡くしているので、次に生まれた男の子はとても大切に育てられた」= His parents lost their first son so they brought him up with extra care, love and attention / His parents lost their first son so they focused all their love, care and attention on their remaining son
「大事に育てられた」は、
"being brought up with care and affection"
"care"は、「細心の注意・気にかけること」
"affection"は、「(子どもなどに対する)愛情」
また、"cherish"「愛情をこめて大事にする」という表現を使うことも出来ます。
"While growing up, He was especially cherished by his parents because of his loss of a sibling."
「彼は兄弟の死により、両親によってとりわけ大切に育てられた。」
"especially"は、「とりわけ・特別に」
"loss of a sibling"は、「兄弟の死」
ご参考になれば幸いです。