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My mistake:D

In Sanskrit, rAdh is a root meaning success, and rAdhA means prosperity among other things. It is used as a name at least since Mahabharata era (final form probably around or slightly later than the beginning of common era), and the popular rAdhA is indeed the "goddess" you refer to. anu- as a prefix has the sense of after or following, so anurAdhA means after rAdhA.

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In astronomy from very long back, the ecliptic was divided into 27 or 28 equal parts, or houses (thus, each part was 13 degrees 20 minutes or about 12 degrees 51 minutes 26 seconds. The significance of the number 27 or 28 is that the moon's siderial period is about 27 d 7 h 43 min 11.6 s, so that on the background of the stars, it advances by about one of these parts each night. (The phase of the moon actually changes slower since the earth is moving too, it takes about 29 d 12 h 44 min 2.8016 s to come back to the same phase.) Each of these houses is named after a prominent asterism in this. rAdhA is one of these houses, and the next house is anurAdhA. The latter asterism corresponds to β, δ, and π, and may also include ν and ρ-Scorpii. I guess this is where the "bright star" meaning comes from.

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The months were named after one of the 12 house closest to the moon during the full moon of the month. This made the Indian year a siderial year (meaning the earth-sun line always pointed to the same place with respect to the stars). When the Indians realized that this differs from the tropical year that controls the seasons because the axis of earth's rotation is not fixed, a correction of a day every 72 years, they played games with all of this siderial system which I do not go into. In any case, the month with full moon near rAdhA (also called vishAkhA, or branched) is around April, or late spring. This might give the association with prosperity.