This one is particularly tricky because what we might think is a verb is actually acting as a noun in this sentence: "unemployment claims."

SUBJECT: 1) A surge (in new home sales) and 2) a drop (in weekly unemployment claims)
VERB: suggest
OBJECT: [the fact] that the economy might not be as weak as some analysts previously thought

The "AND" creates a compound subject, so we must use a plural verbal: "suggest" (or possibly some other plural verb).

A. claims suggest that the economy might not be as weak as some analysts previously thought
No error.

B. claims suggests that the economy might not be so weak as some analysts have previously thought
- "suggests" is singular. Incorrect.
- "so weak as" --> incorrect idiom

C. claims suggest that the economy might not be as weak as have been previously thought by some analysts
- "have been" is incorrect. In this idiom, "the economy might not be as weak as [it] has been thought..." --> "economy" is the implied subject of "has been."
- "as thought by some analysts" is unnecessarily passive. It's better to say "as some analysts thought." (Not grammatically incorrect, but stylistically less preferable).

D. claims, suggesting about the economy that it might not be so weak as previously thought by some analysts
- sentence fragment: there is no main VERB to go with the subject [a surge & a drop].
- "so weak as" --> incorrect idiom
- same passive verb issue.

E. claims, suggesting the economy might not be as weak as previously thought to be by some analysts
- sentence fragment again.
- same passive verb issue.

The correct answer is A.