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Webster's 1828 Dictionary gives these definitions for the verb:
Quote:
1. To fix the mind closely upon a subject; to muse; to dwell upon in thought.
2. To apply the mind to books.
3. To endeavor diligently.
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All three would fit the usage in that passage.
Strong's Concordance gives this definition:
from 4710; to use speed, i.e. to make effort, be prompt or earnest:--do (give) diligence, be diligent (forward), endeavour, labour, study.
From the context, it is obvious that this word means to be diligent in studying out the Bible.